Tootfinder

PDE-aware Optimizer for Physics-informed Neural Networks
Hardik Shukla, Manurag Khullar, Vismay Churiwala
https://arxiv.org/abs/2507.08118 https://arxiv.org/pdf/2507.08118 https://arxiv.org/html/2507.08118
arXiv:2507.08118v1 Announce Type: new
Abstract: Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs) by embedding physical constraints into the loss function. However, standard optimizers such as Adam often struggle to balance competing loss terms, particularly in stiff or ill-conditioned systems. In this work, we propose a PDE-aware optimizer that adapts parameter updates based on the variance of per-sample PDE residual gradients. This method addresses gradient misalignment without incurring the heavy computational costs of second-order optimizers such as SOAP. We benchmark the PDE-aware optimizer against Adam and SOAP on 1D Burgers', Allen-Cahn and Korteweg-de Vries(KdV) equations. Across both PDEs, the PDE-aware optimizer achieves smoother convergence and lower absolute errors, particularly in regions with sharp gradients. Our results demonstrate the effectiveness of PDE residual-aware adaptivity in enhancing stability in PINNs training. While promising, further scaling on larger architectures and hardware accelerators remains an important direction for future research.
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SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning
Qixuan Zhou, Chuqi Chen, Tao Luo, Yang Xiang
https://arxiv.org/abs/2508.10322

SSBE-PINN: A Sobolev Boundary Scheme Boosting Stability and Accuracy in Elliptic/Parabolic PDE Learning
Physics-Informed Neural Networks (PINNs) have emerged as a powerful framework for solving partial differential equations (PDEs), yet they often fail to achieve accurate convergence in the H1 norm, especially in the presence of boundary approximation errors. In this work, we propose a novel method called Sobolev-Stable Boundary Enforcement (SSBE), which redefines the boundary loss using Sobolev norms to incorporate boundary regularity directly into the training process. We provide rigorous theor…

Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
Dominic Blanco, Matthieu Cadiot
https://arxiv.org/abs/2509.10375 h…

Proving symmetry of localized solutions and application to dihedral patterns in the planar Swift-Hohenberg PDE
In this article, we extend the framework developed previously to allow for rigorous proofs of existence of smooth, localized solutions in semi-linear partial differential equations possessing both space and non-space group symmetries. We demonstrate our approach on the Swift-Hohenberg model. In particular, for a given symmetry group $\mathcal{G}$, we construct a natural Hilbert space $H^l_{\mathcal{G}}$ containing only functions with $\mathcal{G}$-symmetry. In this space, products and different…

Neural networks for learning macroscopic chemotactic sensitivity from microscopic models
Radek Erban
https://arxiv.org/abs/2509.12131 https://arxiv.org/pdf…

Neural networks for learning macroscopic chemotactic sensitivity from microscopic models
The macroscopic (population-level) dynamics of chemotactic cell movement -- arising from underlying microscopic (individual-based) models -- are often described by parabolic partial differential equations (PDEs) governing the spatio-temporal evolution of cell concentrations. In certain cases, these macroscopic PDEs can be analytically derived from microscopic models, thereby elucidating the dependence of PDE coefficients on the parameters of the underlying individual-based dynamics. However, su…

The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data
Alina Chertock, Pierre Degond, Amir Sagiv, Li Wang
https://arxiv.org/abs/2507.11399

The Evolution of Pointwise Statistics in Hyperbolic Equations with Random Data
We consider one-dimensional hyperbolic PDEs, linear and nonlinear, with random initial data. Our focus is the {\em pointwise statistics,} i.e., the probability measure of the solution at any fixed point in space and time. For linear hyperbolic equations, the probability density function (PDF) of these statistics satisfies the same linear PDE. For nonlinear hyperbolic PDEs, we derive a linear transport equation for the cumulative distribution function (CDF) and a nonlocal linear PDE for the PDF.…

Pointwise explicit estimates for derivatives of solutions to linear parabolic PDEs with Neumann boundary conditions
C Ciccarella
https://arxiv.org/abs/2507.08622

Pointwise explicit estimates for derivatives of solutions to linear parabolic PDEs with Neumann boundary conditions
We establish a pointwise bound on the spatial derivative $\left| \frac{\partial V}{\partial x} \right|$ of the solution to a parabolic PDE with Neumann boundary conditions. The bound is fully explicit in the sense that it depends only on the coefficients of the PDE and the domain, including closed-form expression for all constants. The proof is purely probabilistic. We first extend to time inhomogeneous diffusions a result concerning the derivative of the solution of a reflected SDE…

Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems
Akshay Thakur, Sawan Kumar, Matthew Zahr, Souvik Chakraborty
https://arxiv.org/abs/2508.09623

Scalable h-adaptive probabilistic solver for time-independent and time-dependent systems
Solving partial differential equations (PDEs) within the framework of probabilistic numerics offers a principled approach to quantifying epistemic uncertainty arising from discretization. By leveraging Gaussian process regression and imposing the governing PDE as a constraint at a finite set of collocation points, probabilistic numerics delivers mesh-free solutions at arbitrary locations. However, the high computational cost, which scales cubically with the number of collocation points, remains…

NeuroPDE: A Neuromorphic PDE Solver Based on Spintronic and Ferroelectric Devices
Siqing Fu, Lizhou Wu, Tiejun Li, Chunyuan Zhang, Sheng Ma, Jianmin Zhang, Yuhan Tang, Jixuan Tang
https://arxiv.org/abs/2507.04677

NeuroPDE: A Neuromorphic PDE Solver Based on Spintronic and Ferroelectric Devices
In recent years, new methods for solving partial differential equations (PDEs) such as Monte Carlo random walk methods have gained considerable attention. However, due to the lack of hardware-intrinsic randomness in the conventional von Neumann architecture, the performance of PDE solvers is limited. In this paper, we introduce NeuroPDE, a hardware design for neuromorphic PDE solvers that utilizes emerging spintronic and ferroelectric devices. NeuroPDE incorporates spin neurons that are capable…

Causal PDE-Control Models: A Structural Framework for Dynamic Portfolio Optimization
Alejandro Rodriguez Dominguez
https://arxiv.org/abs/2509.09585 https://

Causal PDE-Control Models: A Structural Framework for Dynamic Portfolio Optimization
Classical portfolio models collapse under structural breaks, while modern machine-learning allocators adapt flexibly but often at the cost of transparency and interpretability. This paper introduces Causal PDE-Control Models (CPCMs), a unifying framework that integrates causal inference, nonlinear filtering, and forward-backward partial differential equations for dynamic portfolio optimization. The framework delivers three theoretical advances: (i) the existence of conditional risk-neutral meas…

Condition number for finite element discretisation of nonlocal PDE systems with applications to biology
Olusegun E. Adebayo, Raluca Eftimie, Dumitru Trucu
https://arxiv.org/abs/2508.09781

Condition number for finite element discretisation of nonlocal PDE systems with applications to biology
In this work, we investigate the condition number for a system of coupled non-local reaction-diffusion-advection equations developed in the context of modelling normal and abnormal wound healing. Following a finite element discretisation of the coupled non-local system, we establish bounds for this condition number. We further discuss how model parameter choices affect the conditioning of the system. Finally, we discuss how the step size of the chosen time-stepping scheme and the spatial gr…

Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics
Ira J. S. Shokar, Rich R. Kerswell, Peter H. Haynes
https://arxiv.org/abs/2509.09599

Conditioning on PDE Parameters to Generalise Deep Learning Emulation of Stochastic and Chaotic Dynamics
We present a deep learning emulator for stochastic and chaotic spatio-temporal systems, explicitly conditioned on the parameter values of the underlying partial differential equations (PDEs). Our approach involves pre-training the model on a single parameter domain, followed by fine-tuning on a smaller, yet diverse dataset, enabling generalisation across a broad range of parameter values. By incorporating local attention mechanisms, the network is capable of handling varying domain sizes and re…

Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels
Kaijing Lyu, Umberto Biccari, Junmin Wang
https://arxiv.org/abs/2508.03242 https…

Robust stabilization of hyperbolic PDE-ODE systems via Neural Operator-approximated gain kernels
This paper investigates the mean square exponential stabilization problem for a class of coupled PDE-ODE systems with Markov jump parameters. The considered system consists of multiple coupled hyperbolic PDEs and a finite-dimensional ODE, where all system parameters evolve according to a homogeneous continuous-time Markov process. The control design is based on a backstepping approach. To address the computational complexity of solving kernel equations, a DeepONet framework is proposed to learn…

Overcoming the Loss Conditioning Bottleneck in Optimization-Based PDE Solvers: A Novel Well-Conditioned Loss Function
Wenbo Cao, Weiwei Zhang
https://arxiv.org/abs/2508.02692 ht…

Overcoming the Loss Conditioning Bottleneck in Optimization-Based PDE Solvers: A Novel Well-Conditioned Loss Function
Optimization-based PDE solvers that minimize scalar loss functions have gained increasing attention in recent years. These methods either define the loss directly over discrete variables, as in Optimizing a Discrete Loss (ODIL), or indirectly through a neural network surrogate, as in Physics-Informed Neural Networks (PINNs). However, despite their promise, such methods often converge much more slowly than classical iterative solvers and are commonly regarded as inefficient. This work provides a…

Memshare: Memory Sharing for Multicore Computation in R with an Application to Feature Selection by Mutual Information using PDE
Michael C. Thrun, Julian M\"arte
https://arxiv.org/abs/2509.08632

Memshare: Memory Sharing for Multicore Computation in R with an Application to Feature Selection by Mutual Information using PDE
We present memshare\footnote{The Software package is published as a CRAN package under https://CRAN.R-project.org/package=memshare, a package that enables shared memory multicore computation in R by allocating buffers in C++ shared memory and exposing them to R through ALTREP views. We compare memshare to SharedObject (Bioconductor) discuss semantics and safety, and report a 2x speedup over SharedObject with no additional resident memory in a column wise apply benchmark. Finally, we illustrate …

PCGBandit: One-shot acceleration of transient PDE solvers via online-learned preconditioners
Mikhail Khodak, Min Ki Jung, Brian Wynne, Edmond chow, Egemen Kolemen
https://arxiv.org/abs/2509.08765

PCGBandit: One-shot acceleration of transient PDE solvers via online-learned preconditioners
Data-driven acceleration of scientific computing workflows has been a high-profile aim of machine learning (ML) for science, with numerical simulation of transient partial differential equations (PDEs) being one of the main applications. The focus thus far has been on methods that require classical simulations to train, which when combined with the data-hungriness and optimization challenges of neural networks has caused difficulties in demonstrating a convincing advantage against strong classi…

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems
Michael Kartmann, Benedikt Klein, Mario Ohlberger, Thomas Schuster, Stefan Volkwein
https://arxiv.org/abs/2507.11130

Adaptive Reduced Basis Trust Region Methods for Parabolic Inverse Problems
We consider nonlinear inverse problems arising in the context of parameter identification for parabolic partial differential equations (PDEs). For stable reconstructions, regularization methods such as the iteratively regularized Gauss-Newton method (IRGNM) are commonly used, but their application is computationally demanding due to the high-dimensional nature of PDE discretizations. To address this bottleneck, we propose a reduced-order modeling approach that accelerates both the state and adj…

Zeroes of Eigenfunctions of Schr\"odinger Operators after Schwartzman
Willie Wai-Yeung Wong
https://arxiv.org/abs/2509.09739 https://arxiv.org/pdf/250…

Zeroes of Eigenfunctions of Schrödinger Operators after Schwartzman
Consider a complete, connected, smooth, oriented Riemannian manifold $(M,g)$ with boundary, such that the first Betti number vanishes. Sol Schwartzman proved that for Schrödinger operators of the form $-Δ_g + V$ where $\Im(V)$ is signed, if $f: M\to\mathbb{C}$ is a non-vanishing element of its kernel, then $f$ has constant phase. The proof relied on dynamical systems methods applied to the gradient flow of the phase of $f$. In this manuscript we provide a more direct PDE argument that proves …

Permanent Data Encoding (PDE): A Visual Language for Semantic Compression and Knowledge Preservation in 3-Character Units
Yoshiharu Tsuyuki, Xianqi Li, Yuji Kurihara, Kenji Mitsudo
https://arxiv.org/abs/2507.20131

Permanent Data Encoding (PDE): A Visual Language for Semantic Compression and Knowledge Preservation in 3-Character Units
Permanent Data Encoding (PDE) is a visual language framework designed for long-term, human-readable, and electrically independent knowledge preservation. By encoding semantic content into compact 2-3 character alphanumeric codes, paired with public dictionaries and rule-based expansion structures, PDE enables information to be visually interpreted and logically reconstructed without reliance on digital systems. Unlike QR codes or binary data, PDE offers a transparent and self-contained method o…

My talk on "A gradient inequality for continuous functions" at the AMS Sectional meeting in St. Louis is scheduled for Sunday, Oct. 19, 2:45. 🧑‍🏫
#MathConference #RealAnalysis #PDE

Steady advection-diffusion in multiply-connected potential flows
Kyle McKee, Keaton Burns
https://arxiv.org/abs/2509.09444 https://arxiv.org/pdf/2509.09444…

Steady advection-diffusion in multiply-connected potential flows
We consider the steady heat transfer between a collection of impermeable obstacles immersed in an incompressible 2D potential flow, when each obstacle has a prescribed boundary temperature distribution. Inside the fluid, the temperature satisfies a variable-coefficient elliptic partial differential equation (PDE), the solution of which usually requires expensive techniques. To solve this problem efficiently, we construct multiply-connected conformal maps under which both the domain and governin…

Stable and unstable spatially-periodic canards created in singular subcritical Turing bifurcations in the Brusselator system
Robert Jencks, Arjen Doelman, Tasso J. Kaper, Theodore Vo
https://arxiv.org/abs/2509.04835

Stable and unstable spatially-periodic canards created in singular subcritical Turing bifurcations in the Brusselator system
In this article, we study the Brusselator partial differential equation (PDE) in the limit in which the diffusivity of the activator is much smaller than that of the inhibitor. The PDE robustly exhibits a subcritical Turing bifurcation that, in this limit, is labeled as a singular Turing bifurcation. We show that families of spatially-periodic canard solutions emerge from this subcritical singular Turing bifurcation. Then, right after they emerge, the solutions lose their purely sinusoidal stru…

The Schwarz lemma for holomorphic and minimal disks at the boundary
David Kalaj
https://arxiv.org/abs/2509.09471 https://arxiv.org/pdf/2509.09471

The Schwarz lemma for holomorphic and minimal disks at the boundary
We first prove a Boundary Schwarz lemma for holomorphic disks on the unit ball in $\mathbb{C}^n$. Further by using a Schwarz lemma for minimal conformal disks of Forstneri\v c and Kalaj (F.~Forstneri{č} and D.~Kalaj. \newblock Schwarz-pick lemma for harmonic maps which are conformal at a point. \newblock {\em Anal. PDE}, 17(3):981--1003, 2024.) we prove the boundary Schwarz lemma for such minimal disks.

Control-affine Schr\"odinger Bridge and Generalized Bohm Potential
Alexis M. H. Teter, Abhishek Halder, Michael D. Schneider, Alexx S. Perloff, Jane Pratt, Conor M. Artman, Maria Demireva
https://arxiv.org/abs/2508.08511

Control-affine Schrödinger Bridge and Generalized Bohm Potential
The control-affine Schrödinger bridge concerns with a stochastic optimal control problem. Its solution is a controlled evolution of joint state probability density subject to a control-affine Itô diffusion with a given deadline connecting a given pair of initial and terminal densities. In this work, we recast the necessary conditions of optimality for the control-affine Schrödinger bridge problem as a two point boundary value problem for a quantum mechanical Schrödinger PDE with complex pot…

Bridging Sequential Deep Operator Network and Video Diffusion: Residual Refinement of Spatio-Temporal PDE Solutions
Jaewan Park, Farid Ahmed, Kazuma Kobayashi, Seid Koric, Syed Bahauddin Alam, Iwona Jasiuk, Diab Abueidda
https://arxiv.org/abs/2507.06133

Bridging Sequential Deep Operator Network and Video Diffusion: Residual Refinement of Spatio-Temporal PDE Solutions
Video-diffusion models have recently set the standard in video generation, inpainting, and domain translation thanks to their training stability and high perceptual fidelity. Building on these strengths, we repurpose conditional video diffusion as a physics surrogate for spatio-temporal fields governed by partial differential equations (PDEs). Our two-stage surrogate first applies a Sequential Deep Operator Network (S-DeepONet) to produce a coarse, physics-consistent prior from the prescribed b…

Edwards-Wilkinson limit for a stochastic advection-diffusion PDE
Sotirios Kotitsas, Dejun Luo, Mario Maurelli
https://arxiv.org/abs/2509.07956 https://arxi…

Edwards-Wilkinson limit for a stochastic advection-diffusion PDE
We consider a diffusion in a Gaussian random environment that is white in time and study the large-scale behavior of the quenched density with respect to the Lebesgue measure. We show that under diffusive rescaling, the fluctuations of the density converge to a Gaussian limit, described by an additive stochastic heat equation. In the case where the environment is divergence-free, our result can be interpreted as computing the scaling limit of the first-order correction to the quenched Central L…

Beyond Blur: A Fluid Perspective on Generative Diffusion Models
Grzegorz Gruszczynski, Michal Jan Wlodarczyk, Jakub J Meixner, Przemyslaw Musialski
https://arxiv.org/abs/2506.16827

Beyond Blur: A Fluid Perspective on Generative Diffusion Models
We propose a novel PDE-driven corruption process for generative image synthesis based on advection-diffusion processes which generalizes existing PDE-based approaches. Our forward pass formulates image corruption via a physically motivated PDE that couples directional advection with isotropic diffusion and Gaussian noise, controlled by dimensionless numbers (Peclet, Fourier). We implement this PDE numerically through a GPU-accelerated custom Lattice Boltzmann solver for fast evaluation. To indu…

A note on a diffeomorphism criterion via long-time Ricci flow
Shaochuang Huang, Zhuo Peng
https://arxiv.org/abs/2509.05802 https://arxiv.org/pdf/2509.05802…

A note on a diffeomorphism criterion via long-time Ricci flow
In this note, we give a diffeomorphism (to $\mathbb{R}^n$) criterion via long-time Ricci flow and show some applications. In particular, we provide an affirmative answer that the conclusion in [Manifolds with small curvature concentration, Ann. PDE, 2024] by Chan, Lee and the first named author and [Removing scalar curvature assumption for Ricci flow smoothing, Bull. Lond. Math. Soc., 2025] by A. Martens about manifolds with small curvature concentration can be improved to diffeomorphism in dim…

Mapping the Photon Detection Efficiency of VUV Sensitive SiPMs from the Ultra-Violet to the Near Infra-Red
Austin de St Croix, Harry Lewis, Kurtis Raymond, Fabrice Reti\`ere, Maia Henriksson-Ward, Giacomo Gallina, Nicholas Morrison, Aileen Zhang
https://arxiv.org/abs/2508.16005

Mapping the Photon Detection Efficiency of VUV Sensitive SiPMs from the Ultra-Violet to the Near Infra-Red
We present a flexible analytic model describing the Photo-Detection Efficiency (PDE) for P-on-N silicon photomultipliers. This model can be used for device characterization, PDE extrapolation from limited data or design optimization. The model uses six device specific parameters to describe PDE for a large range of wavelengths, angles of incidence, voltages and temperatures. Refractive index data taken from literature are inputs to the model, which also allows the prediction of PDE for devices …

Silicon single-photon detector achieving over 84% photon detection efficiency with flexible operation modes
Dong An, Chao Yu, Ming-Yang Zheng, Anran Guo, Junsong Wang, Ruizhi Li, Huaping Ma, Xiu-Ping Xie, Xiao-Hui Bao, Qiang Zhang, Jun Zhang, Jian-Wei Pan
https://arxiv.org/abs/2507.18172

Silicon single-photon detector achieving over 84% photon detection efficiency with flexible operation modes
Silicon single-photon detectors (Si SPDs) play a crucial role in detecting single photons in the visible spectrum. For various applications, photon detection efficiency (PDE) is the most critical characteristic for effectively collecting photons. Here, we present a Si SPD with a remarkable PDE of up to 84.4% at 785 nm, supporting multiple operation modes. We design and fabricate a thick-junction Si single-photon avalanche diode (SPAD) that enhances the avalanche probability through a backside-i…

from my link log —
The (unfinished) partial differential equation coffee table book.
https://people.maths.ox.ac.uk/trefethen/pdectb.html
saved 2025-07-17

Generalized Plane Wave quasi-Trefftz spaces for wave propagation in inhomogeneous media
Ilaria Fontana, Lise-Marie Imbert-Gerard
https://arxiv.org/abs/2508.09435 https://…

Generalized Plane Wave quasi-Trefftz spaces for wave propagation in inhomogeneous media
Partial Differential Equations (PDEs) models for wave propagation in inhomogeneous media are relevant for many applications. We will discuss numerical methods tailored for tackling problems governed by these variable-coefficient PDEs. Trefftz methods rely, in broad terms, on the idea of approximating solutions to PDEs via Galerkin methods using basis functions that are exact solutions of the PDE, making explicit use of information about the ambient medium. However, wave propagation in inhomogen…

PDE-constrained optimal control of a leader-follower opinion formation model
Bertram D\"uring, Oliver Wright
https://arxiv.org/abs/2508.21674 https://…

PDE-constrained optimal control of a leader-follower opinion formation model
We consider the PDE-constrained optimal control of a leader-follower kinetic opinion formation model, with a Fokker-Planck-type system of partial differential equations as a state constraint. We derive the Boltzmann-type and Fokker-Planck-type systems of equations associated with the controlled leader-follower opinion formation model. In a function space setting we derive first-order optimality conditions associated with the PDE-constrained optimal control problem, yielding an optimality system…

Functional analysis and partial differential equations in spectral Barron spaces
Mourad Choulli, Shuai Lu, Hiroshi Takase
https://arxiv.org/abs/2507.06778 …

Functional analysis and partial differential equations in spectral Barron spaces
Spectral Barron spaces, constituting a specialized class of function spaces that serve as an interdisciplinary bridge between mathematical analysis, partial differential equations (PDEs), and machine learning, are distinguished by the decay profiles of their Fourier transform. In this work, we shift from conventional numerical approximation frameworks to explore advanced functional analysis and PDE theoretic perspectives within these spaces. Specifically, we present a rigorous characterization …

Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia
Ivan Yevenko, Hiroki Kojima, Chrystopher L. Nehaniv
https://arxiv.org/abs/2508.02935 https://

Using Dynamical Systems Theory to Quantify Complexity in Asymptotic Lenia
Continuous cellular automata (CCAs) have evolved from discrete lookup tables to continuous partial differential equation (PDE) formulations in the search for novel forms of complexity. Despite innovations in qualitative behavior, analytical methods have lagged behind, reinforcing the notion that emergent complexity defies simple explanation. In this paper, we demonstrate that the PDE formulation of Asymptotic Lenia enables rigorous analysis using dynamical systems theory. We apply the concepts …

Evaluating PDE discovery methods for multiscale modeling of biological signals
Andr\'ea Ducos (AISTROSIGHT), Audrey Denizot (AISTROSIGHT), Thomas Guyet (AISTROSIGHT), Hugues Berry (AISTROSIGHT)
https://arxiv.org/abs/2506.20694

Evaluating PDE discovery methods for multiscale modeling of biological signals
Biological systems are non-linear, include unobserved variables and the physical principles that govern their dynamics are partly unknown. This makes the characterization of their behavior very challenging. Notably, their activity occurs on multiple interdependent spatial and temporal scales that require linking mechanisms across scales. To address the challenge of bridging gaps between scales, we leverage partial differential equations (PDE) discovery. PDE discovery suggests meso-scale dynamic…

LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process
Xiaodong Feng, Ling Guo, Xiaoliang Wan, Hao Wu, Tao Zhou, Wenwen Zhou
https://arxiv.org/abs/2507.22493

LVM-GP: Uncertainty-Aware PDE Solver via coupling latent variable model and Gaussian process
We propose a novel probabilistic framework, termed LVM-GP, for uncertainty quantification in solving forward and inverse partial differential equations (PDEs) with noisy data. The core idea is to construct a stochastic mapping from the input to a high-dimensional latent representation, enabling uncertainty-aware prediction of the solution. Specifically, the architecture consists of a confidence-aware encoder and a probabilistic decoder. The encoder implements a high-dimensional latent variable …

PDE-Constrained High-Order Mesh Optimization
Tzanio Kolev, Boyan Lazarov, Ketan Mittal, Mathias Schmidt, Vladimir Tomov
https://arxiv.org/abs/2507.01917 ht…

PDE-Constrained High-Order Mesh Optimization
We present a novel framework for PDE-constrained $r$-adaptivity of high-order meshes. The proposed method formulates mesh movement as an optimization problem, with an objective function defined as a convex combination of a mesh quality metric and a measure of the accuracy of the PDE solution obtained via finite element discretization. The proposed formulation achieves optimized, well-defined high-order meshes by integrating mesh quality control, PDE solution accuracy, and robust gradient regula…

BiLO: Bilevel Local Operator Learning for PDE Inverse Problems. Part II: Efficient Uncertainty Quantification with Low-Rank Adaptation
Ray Zirui Zhang, Christopher E. Miles, Xiaohui Xie, John S. Lowengrub
https://arxiv.org/abs/2507.17019

BiLO: Bilevel Local Operator Learning for PDE Inverse Problems. Part II: Efficient Uncertainty Quantification with Low-Rank Adaptation
Uncertainty quantification and inverse problems governed by partial differential equations (PDEs) are central to a wide range of scientific and engineering applications. In this second part of a two part series, we extend Bilevel Local Operator Learning (BiLO) for PDE-constrained optimization problems developed in Part 1 to the Bayesian inference framework. At the lower level, we train a network to approximate the local solution operator by minimizing the local operator loss with respect to the…

A nonstandard finite difference scheme for an SEIQR epidemiological PDE mode
Achraf Zinihi, Matthias Ehrhardt, Moulay Rchid Sidi Ammi
https://arxiv.org/abs/2508.02928 https://…

A nonstandard finite difference scheme for an SEIQR epidemiological PDE mode
This paper introduces a nonstandard finite difference (NSFD) approach to a reaction-diffusion SEIQR epidemiological model, which captures the spatiotemporal dynamics of infectious disease transmission. Formulated as a system of semilinear parabolic partial differential equations (PDEs), the model extends classical compartmental models by incorporating spatial diffusion to account for population movement and spatial heterogeneity. The proposed NSFD discretization is designed to preserve the cont…

Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems
Yu-Sen Yang, Ling Guo, Xiaodan Ren
https://arxiv.org/abs/2507.19888 https://

Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems
Multi-scale PDE problems present significant challenges in scientific computing. While conventional MLP-based deep learning methods exhibit spectral bias in resolving multi-scale features, the physics-informed Kolmogorov-Arnold network (PIKAN) mitigates this issue through its novel architecture, demonstrating certain advantages. On the other hand, insights from the information bottleneck theory suggest that high-resolution training points are essential for these hybrid methods to accurately cap…

Rearrangements and infimum convolutions
Devraj Duggal, James Melbourne, Cyril Roberto
https://arxiv.org/abs/2508.07983 https://arxiv.org/pdf/2508.07983

Rearrangements and infimum convolutions
We develop a general comparison result for inf convolution operators related to rearrangement. As a consequence we derive comparison results under spherically symmetric rearrangement for Laplace and polar transforms. As a by product we simplify existing proofs related to the functional Blaschke Santalo inequality of Keith Ball and derive a comparison result for some parabolic PDE.

Fast prediction of the hydrodynamic QGP evolution in ultra-relativistic heavy-ion collisions using Fourier Neural Operators
David Stewart, Joern Putschke
https://arxiv.org/abs/2507.23598

Fast prediction of the hydrodynamic QGP evolution in ultra-relativistic heavy-ion collisions using Fourier Neural Operators
Recent research in machine learning has employed neural networks to learn mappings between function spaces on bounded domains termed ``neural operators''. As such, these operators can provide alternatives to standard numerical methods for partial differential equation (PDE) solutions. In particular, the Fourier Neural Operator (FNO) has been shown to map solutions for classical fluid flow problems with accuracy competitive with traditional PDE solvers and with much greater computing speed. This…

A novel approach to study the wellposedness of the 3D fluid-2D plate interaction PDE System
George Avalos, Pelin G. Geredeli, Hemanta Kunwar, Hyesuk Lee
https://arxiv.org/abs/2509.03431

A novel approach to study the wellposedness of the 3D fluid-2D plate interaction PDE System
We consider a certain fluid-structure interaction (FSI) system with a view of obtaining an alternative methodology for establishing its strongly continuous semigroup wellposedness. (Semigroup generation for this FSI was originally considered in Avalos-Clark (2014).) The FSI model under consideration describes the vibrations of an incompressible fluid within a 3D cavity as it interacts with the elastic membrane on the ``free" upper boundary of the cavity. Such coupled PDE systems appear in varie…

Multi-stage PDE-based image processing techniques for noisy MRI scans
Ksenia Slepova, Ivan Etoku Oiye, Martin B. van Gijzen
https://arxiv.org/abs/2509.02342 https://

Multi-stage PDE-based image processing techniques for noisy MRI scans
Image denoising and image segmentation play essential roles in image processing. Partial differential equations (PDE)-based methods have proven to show reliable results when incorporated in both denoising and segmentation of images. In our work, we discuss a multi-stage PDE-based image processing approach. It relies upon the nonlinear diffusion for noise removal and clustering and region growing for segmentation. In the first stage of the approach, the raw image is computed from noisy measureme…

First-order continuum models for nonlinear dispersive waves in the granular crystal lattice
Su Yang, Gino Biondini, Christopher Chong, Panayotis G. Kevrekidis
https://arxiv.org/abs/2507.07571

First-order continuum models for nonlinear dispersive waves in the granular crystal lattice
We derive and analyze, analytically and numerically, two first-order continuum models to approximate the nonlinear dynamics of granular crystal lattices, focusing specifically on solitary waves, periodic waves, and dispersive shock waves. The dispersive shock waves predicted by the two continuum models are studied using modulation theory, DSW fitting techniques, and direct numerical simulations. The PDE-based predictions show good agreement with the DSWs generated by the discrete model simulati…

Analysis and Numerical Approximation to Interactive Dynamics of Navier Stokes-Plate Interaction PDE System
Pelin G. Geredeli (Clemson University), Quyuan Lin (Clemson University), Dylan Mcknight (Colorado Mesa University), Mohammad Mahabubur Rahman (Clemson University)
https://arxiv.org/abs/2507.02230

Analysis and Numerical Approximation to Interactive Dynamics of Navier Stokes-Plate Interaction PDE System
We consider a Navier-Stokes fluid-plate interaction (FSI) system which describes the evolutions of the fluid contained within a 3D cavity, as it interacts with a deformable elastic membrane on the ``free" upper boundary of the cavity. These models arise in various aeroelastic and biomedical applications as well as in the control of ocular pressure, and sloshing phenomena. We analyze the well-posedness of weak solutions to the stationary ($λ$-parametrized) coupled PDE system by way of invoking …

HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions
Rafael Bischof, Michal Piovar\v{c}i, Michael A. Kraus, Siddhartha Mishra, Bernd Bickel
https://arxiv.org/abs/2509.05117

HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions
We present HyPINO, a multi-physics neural operator designed for zero-shot generalization across a broad class of parametric PDEs without requiring task-specific fine-tuning. Our approach combines a Swin Transformer-based hypernetwork with mixed supervision: (i) labeled data from analytical solutions generated via the Method of Manufactured Solutions (MMS), and (ii) unlabeled samples optimized using physics-informed objectives. The model maps PDE parametrizations to target Physics-Informed Neura…

Numerical PDE solvers outperform neural PDE solvers
Patrick Chatain, Michael Rizvi-Martel, Guillaume Rabusseau, Adam Oberman
https://arxiv.org/abs/2507.21269 https://

Numerical PDE solvers outperform neural PDE solvers
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional architecture, DeepFDM enforces stability and first-order convergence via CFL-compliant coefficient parameterizations. Model weights correspond directly to PDE coefficients, yielding an interpretable inverse-problem formulation. We evaluate DeepFDM on a bench…

Accurate and scalable deep Maxwell solvers using multilevel iterative methods
Chenkai Mao, Jonathan A. Fan
https://arxiv.org/abs/2509.03622 https://arxiv.o…

Accurate and scalable deep Maxwell solvers using multilevel iterative methods
Neural networks have promise as surrogate partial differential equation (PDE) solvers, but it remains a challenge to use these concepts to solve problems with high accuracy and scalability. In this work, we show that neural network surrogates can combine with iterative algorithms to accurately solve PDE problems featuring different scales, resolutions, and boundary conditions. We develop a subdomain neural operator model that supports arbitrary Robin-type boundary condition inputs, and we show …

Multi-species McKean-Vlasov dynamics in non-convex landscapes
Manh Hong Duong, Grigorios A. Pavliotis, Julian Tugaut
https://arxiv.org/abs/2507.07617 https…

Multi-species McKean-Vlasov dynamics in non-convex landscapes
In this paper, we study multi-species stochastic interacting particle systems and their mean-field McKean-Vlasov partial differential equations (PDEs) in non-convex landscapes. We discuss the well-posedness of the multi-species SDE system, propagation of chaos and the derivation of the coupled McKean-Vlasov PDE system in the mean field limit. Our focus is on the long-time and asymptotic behaviour of the mean-field PDEs. Under suitable growth assumptions on the potentials and an appropriate stru…

Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass
Jeongheon Park, Soonsik Kwon, Taegyu Kim
https://arxiv.org/abs/2508.07787

Blow-up construction and instability for mass-critical half-wave equation with slightly superthreshold mass
We study the blow-up dynamics for the $L^2$-critical focusing half-wave equation on the real line, a nonlocal dispersive PDE arising in various physical models. As in other mass-critical models, the ground state solution becomes a threshold between the global well-posedness and the existence of a blow-up. The first blow-up construction is due to Krieger, Lenzmann and Raphaël, in which they constructed the minimal mass blow-up solution at the threshold mass. In this paper, we construct finite-t…

Boundary behavior in the Loewner-Nirenberg problem
Satyanad Kichenassamy (LMR)
https://arxiv.org/abs/2507.02484 https://arxiv.org/pdf…

Boundary behavior in the Loewner-Nirenberg problem
Let $Ω\subset\mathbb R^n$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $n\geq 3$ and $u_Ω$ is the maximal solution of equation $Δu = n(n-2)u^{(n+2)/(n-2)}$ in $Ω$, then the hyperbolic radius $v_Ω=u_Ω^{-2/(n-2)}$ is of class $C^{2+α}$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE.

Goal-oriented optimal sensor placement for PDE-constrained inverse problems in crisis management
Marco Mattuschka, Noah An der Lan, Max von Danwitz, Daniel Wolff, Alexander Popp
https://arxiv.org/abs/2507.02500

Goal-oriented optimal sensor placement for PDE-constrained inverse problems in crisis management
This paper presents a novel framework for goal-oriented optimal static sensor placement and dynamic sensor steering in PDE-constrained inverse problems, utilizing a Bayesian approach accelerated by low-rank approximations. The framework is applied to airborne contaminant tracking, extending recent dynamic sensor steering methods to complex geometries for computational efficiency. A C-optimal design criterion is employed to strategically place sensors, minimizing uncertainty in predictions. Nume…

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates
Andrew Hassell, Qiuye Jia, Ethan Sussman, Andras Vasy
https://arxiv.org/abs/2509.09518

Microlocal analysis of the non-relativistic limit of the Klein--Gordon equation: Estimates
This is the more technical half of a two-part work in which we introduce a robust microlocal framework for analyzing the non-relativistic limit of relativistic wave equations with time-dependent coefficients, focusing on the Klein--Gordon equation. Two asymptotic regimes in phase space are relevant to the non-relativistic limit: one corresponding to what physicists call ``natural'' units, in which the PDE is approximable by the free Klein--Gordon equation, and a low-frequency regime in which th…

Replaced article(s) found for cs.LG. https://arxiv.org/list/cs.LG/new
[3/5]:
- Towards Reasoning for PDE Foundation Models: A Reward-Model-Driven Inference-Time-Scaling Algorithm
Siddharth Mansingh, et al.

Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$
Guanhua Shao, Jiahua Zou
https://arxiv.org/abs/2507.18825 https://a…

Self-shrinkers with any number of ends in $\mathbb{R}^{3}$ by stacking $\mathbb{R}^{2}$
For each half-integer $J$ and large enough integer $m$ we construct by PDE gluing methods a self-shrinker $\breve{M}[J,m]$ with $2J+1$ ends and genus $2J(m-1)$. $\breve{M}[J,m]$ resembles the stacking of $2J+1$ levels of the plane $\mathbb{R}^2$ in $\mathbb{R}^3$ that have been connected by $2Jm$ catenoidal bridges with $m$ bridges connecting each pair of adjacent levels. It observes the symmetry of an $m$-gonal prism (when $J$ is a half integer) or an $m$-gonal antiprism (when $J$ is an intege…

A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis
Iulian C\^impean, Andreea Grecu, Liviu Marin
https://arxiv.org/abs/2508.08215

A probabilistic approach to spectral analysis of Cauchy-type inverse problems: Convergence and stability analysis
A comprehensive convergence and stability analysis of some probabilistic numerical methods designed to solve Cauchy-type inverse problems is performed in this study. Such inverse problems aim at solving an elliptic partial differential equation (PDE) or a system of elliptic PDEs in a bounded Euclidean domain, subject to incomplete boundary and/or internal conditions, and are usually severely ill-posed. In a very recent paper \cite{CiGrMaI}, a probabilistic numerical framework has been developed…

Causal Multi-fidelity Surrogate Forward and Inverse Models for ICF Implosions
Tyler E. Maltba, Ben S. Southworth, Jeffrey R. Haack, Marc L. Klasky
https://arxiv.org/abs/2509.05510

Causal Multi-fidelity Surrogate Forward and Inverse Models for ICF Implosions
Continued progress in inertial confinement fusion (ICF) requires solving inverse problems relating experimental observations to simulation input parameters, followed by design optimization. However, such high dimensional dynamic PDE-constrained optimization problems are extremely challenging or even intractable. It has been recently shown that inverse problems can be solved by only considering certain robust features. Here we consider the ICF capsule's deuterium-tritium (DT) interface, and cons…

MNO : A Multi-modal Neural Operator for Parametric Nonlinear BVPs
Vamshi C. Madala, Nithin Govindarajan, Shivkumar Chandrasekaran
https://arxiv.org/abs/2507.11870

MNO : A Multi-modal Neural Operator for Parametric Nonlinear BVPs
We introduce a novel Multimodal Neural Operator (MNO) architecture designed to learn solution operators for multi-parameter nonlinear boundary value problems (BVPs). Traditional neural operators primarily map either the PDE coefficients or source terms independently to the solution, limiting their flexibility and applicability. In contrast, our proposed MNO architecture generalizes these approaches by mapping multiple parameters including PDE coefficients, source terms, and boundary conditions …

Optimal control of SDEs with merely measurable drift: an HJB approach
Kai Du, Qingmeng Wei
https://arxiv.org/abs/2509.01054 https://arxiv.org/pdf/2509.0105…

Optimal control of SDEs with merely measurable drift: an HJB approach
We investigate an optimal control problem for a diffusion whose drift and running cost are merely measurable in the state variable. Such low regularity rules out the use of Pontryagin's maximum principle and also invalidates the standard proof of the Bellman principle of optimality. We address these difficulties by analyzing the associated Hamilton-Jacobi-Bellman (HJB) equation. Using PDE techniques together with a policy iteration scheme, we prove that the HJB equation admits a unique strong s…

Sequential Neural Operator Transformer for High-Fidelity Surrogates of Time-Dependent Non-linear Partial Differential Equations
Qibang Liu, Seid Koric
https://arxiv.org/abs/2507.03272

Sequential Neural Operator Transformer for High-Fidelity Surrogates of Time-Dependent Non-linear Partial Differential Equations
Partial differential equations (PDEs) are fundamental to modeling complex and nonlinear physical phenomena, but their numerical solution often requires significant computational resources, particularly when a large number of forward full solution evaluations are necessary, such as in design, optimization, sensitivity analysis, and uncertainty quantification. Recent progress in operator learning has enabled surrogate models that efficiently predict full PDE solution fields; however, these models…

Curvature Flow of Networks with Triple Junction Drag and Grain Rotation
Yuchuan Yang, Selim Esedoglu
https://arxiv.org/abs/2509.06125 https://arxiv.org/pdf…

Curvature Flow of Networks with Triple Junction Drag and Grain Rotation
We prove a local existence result for a PDE system that describes curvature motion of networks with a dynamic boundary condition known as triple junction drag. This model arises in the study of grain boundary evolution in polycrystalline materials. We demonstrate the possibility of a new type of topological change during the evolution of the network. Moreover, we adopt a dynamic surface tension model that depends on the crystallographic orientations of the grains, which are allowed to rotate. L…

A surface finite element scheme for a stochastic PDE on an evolving curve
Paola Pozzi, Bj\"orn Stinner
https://arxiv.org/abs/2507.01527 https://

A surface finite element scheme for a stochastic PDE on an evolving curve
In this paper we consider an ESFEM method for the advection and diffusion of a scalar quantity on a moving closed curve. The diffusion process is controlled by a forcing term that may include a rough term (specifically a stochastic noise) which in particular destroys the classical time differentiability properties of the solution. We provide a suitable variational solution concept and a fully discrete FEM discretization. Our error analysis appropriately generalizes classical estimates to this w…

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs
Dante Kalise, Lucas M. Moschen, Grigorios A. Pavliotis
https://arxiv.org/abs/2507.12411 ht…

Linearization-Based Feedback Stabilization of McKean-Vlasov PDEs
We study the feedback stabilization of the McKean-Vlasov PDE on the torus. Our goal is to steer the dynamics toward a prescribed stationary distribution or accelerate convergence to it using a time-dependent control potential. We reformulate the controlled PDE in a weighted-projected space and apply the ground-state transform to obtain a Schrodinger-type operator. The resulting operator framework enables spectral analysis, verification of the infinite-dimensional Hautus test, and the constructi…

Boundary Feedback and Observer Synthesis for a Class of Nonlinear Parabolic--Elliptic PDE Systems
Kamal Fenza, Moussa Labbadi, Mohamed Ouzahra
https://arxiv.org/abs/2507.12615

Boundary Feedback and Observer Synthesis for a Class of Nonlinear Parabolic--Elliptic PDE Systems
This paper investigates the stabilization of a coupled system comprising a parabolic PDE and an elliptic PDE with nonlinear terms. A rigorous backstepping design provides an explicit boundary control law and exponentially convergent observers from partial boundary measurements. Several theorems ensure exponential stability and well-posedness of the nonlinear closed-loop system.

Kernel-based Greedy Approximation of Parametric Elliptic Boundary Value Problems
Bernard Haasdonk, Gabriele Santin, Tizian Wenzel
https://arxiv.org/abs/2507.06731

Kernel-based Greedy Approximation of Parametric Elliptic Boundary Value Problems
We recently introduced a scale of kernel-based greedy schemes for approximating the solutions of elliptic boundary value problems. The procedure is based on a generalized interpolation framework in reproducing kernel Hilbert spaces and was coined PDE-$β$-greedy procedure, where the parameter $β\geq 0$ is used in a greedy selection criterion and steers the degree of function adaptivity. Algebraic convergence rates have been obtained for Sobolev-space kernels and solutions of finite smoothness.…

The $L^p$ regularity problem for parabolic operators with transversally independent coefficients
Martin Dindo\v{s}, Jill Pipher, Martin Ulmer
https://arxiv.org/abs/2509.06627 ht…

The $L^p$ regularity problem for parabolic operators with transversally independent coefficients
In this paper, we fully resolve the question of whether the Regularity problem for the parabolic PDE $\partial_tu - \mbox{div}(A\nabla u)=0$ on the domain $\mathbb R^{n+1}_+\times\mathbb R$ is solvable for some $p\in (1,\infty)$ under the assumption that the matrix $A$ is elliptic, has bounded and measurable coefficients and its coefficients are independent of the spatial variable $x_{n+1}$ (which is transversal to the boundary). We prove that for some $p_0>1$ the Regularity problem is solvable…

Stabilizing PDE--ML Coupled System
Saad Qadeer, Panos Stinis, Hui Wan
https://arxiv.org/abs/2506.19274 https://arxiv.org/pdf/2506.192…

Stabilizing PDE--ML Coupled System
A long-standing obstacle in the use of machine-learnt surrogates with larger PDE systems is the onset of instabilities when solved numerically. Efforts towards ameliorating these have mostly concentrated on improving the accuracy of the surrogates or imbuing them with additional structure, and have garnered limited success. In this article, we study a prototype problem and draw insights that can help with more complex systems. In particular, we focus on a viscous Burgers'-ML system and, after i…

Crosslisted article(s) found for math.PR. https://arxiv.org/list/math.PR/new
[2/2]:
- Is RL fine-tuning harder than regression? A PDE learning approach for diffusion models
Wenlong Mou

Sensor placement via large deviations in the Eikonal equation
Ilias Ftouhi, Enrique Zuazua
https://arxiv.org/abs/2508.21469 https://arxiv.org/pdf/2508.2146…

Sensor placement via large deviations in the Eikonal equation
In this work, we address the problem of optimally placing a finite number of sensors within a given region so as to minimize the mean or maximal distance to the points of the domain. To tackle this natural geometric performance criterion, formulated in terms of distance functions, we combine tools from geometric analysis with a classical result of Varadhan, which provides an efficient approximation of the distance function via the solution of a simple elliptic PDE. The effectiveness of the prop…

On the Gromov--Hausdorff stability of metric viscosity solutions
Shimpei Makida
https://arxiv.org/abs/2507.06495 https://arxiv.org/pdf/2507.06495 https://arxiv.org/html/2507.06495
arXiv:2507.06495v1 Announce Type: new
Abstract: We establish the stability of metric viscosity solutions to first-order Hamilton--Jacobi equations under Gromov--Hausdorff convergence. Our proof combines a characterization of metric viscosity solutions via quadratic distance functions with a doubling variable method adapted to epsilon-isometries, which allows us to pass to the Gromov--Hausdorff limit without embedding the spaces into a common ambient space. As a byproduct, we give a PDE-based proof of the stability of the dual Kantorovich problems under measured-Gromov--Hausdorff convergence.
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Physics-informed low-rank neural operators with application to parametric elliptic PDEs
Sebastian Schaffer, Lukas Exl
https://arxiv.org/abs/2509.07687 https://

Physics-informed low-rank neural operators with application to parametric elliptic PDEs
We present the Physics-Informed Low-Rank Neural Operator (PILNO), a neural operator framework for efficiently approximating solution operators of partial differential equations (PDEs) on point cloud data. PILNO combines low-rank kernel approximations with an encoder--decoder architecture, enabling fast, continuous one-shot predictions while remaining independent of specific discretizations. The model is trained using a physics-informed penalty framework, ensuring that PDE constraints and bounda…

Analysis of Reaction-Diffusion Predator-Prey System under Random Switching
Nguyen H. Du, Nhu N. Nguyen
https://arxiv.org/abs/2507.06491 https://arxiv.org/pdf/2507.06491 https://arxiv.org/html/2507.06491
arXiv:2507.06491v1 Announce Type: new
Abstract: This paper investigates the long-term dynamics of a reaction-diffusion predator-prey system subject to random environmental fluctuations modeled by Markovian switching. The model is formulated as a hybrid system of partial differential equations (PDEs), where the switching between different ecological regimes captures the randomness in environmental conditions. We derive a critical threshold parameter that determines whether the predator species will eventually go extinct or persist. We further characterize the system's asymptotic behavior by providing a detailed pathwise description of the omega-limit set of solutions. This analysis reveals how the effects of random switching shape the distribution and long-term coexistence of the species. Numerical simulations are provided to validate and illustrate the theoretical findings, highlighting transitions between different dynamical regimes. To the best of our knowledge, this is the first work that rigorously analyzes a spatially diffusive predator-prey model under Markovian switching, thereby bridging the gap between spatial ecology and stochastic hybrid PDE systems.
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Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential
Alekos Cecchin, Paul Nikolaev
https://arxiv.org/abs/2509.01266

Convergence rate for Fluctuations of mean field interacting diffusion and application to 2D viscous Vortex model and Coulomb potential
For a system of mean field interacting diffusion on $\mathbb{T}^d$, the empirical measure $μ^N$ converges to the solution $μ$ of the Fokker-Planck equation. Refining this mean field limit as a Central Limit Theorem, the fluctuation process $ρ^N_t= \sqrt{N}( μ^N_t -μ_t)$ convergences to the solution $ρ$ of a linear stochastic PDE on the negative Sobolev space $H^{-λ-2}(\mathbb{T}^d)$. The main result of the paper is to establish a rate for such convergence: we show that $|\mathbb{E}[…

Robust PDE discovery under sparse and highly noisy conditions via attention neural networks
Shilin Zhang, Yunqing Huang, Nianyu Yi, shihan Zhang
https://arxiv.org/abs/2506.17908

Robust PDE discovery under sparse and highly noisy conditions via attention neural networks
The discovery of partial differential equations (PDEs) from experimental data holds great promise for uncovering predictive models of complex physical systems. In this study, we introduce an efficient automatic model discovery framework, ANN-PYSR, which integrates attention neural networks with the state-of-the-art PySR symbolic regression library. Our approach successfully identifies the governing PDE in six benchmark examples. Compared to the DLGA framework, numerical experiments demonstrate …

Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D
Aussie Greene, Larry L. Schumaker
https://arxiv.org/abs/2508.16060 https://

Using the Immersed Penalized Boundary Method with Splines to Solve PDE's on Curved Domains in 3D
Second-order elliptic boundary-value problems defined on curved domains in 2D and 3D arise frequently in practice. A lot of work has gone into developing numerical methods for solving such problems. One of the newest and most promising methods is the $\textit{immersed penalized boundary method}$ (IPBM) introduced in [Schumaker, L. L., Solving elliptic PDE's on domains with curved boundaries with an immersed penalized boundary method, J. Sci. Comp. ${\bf 80(3)}$ (2019), 1369--1394]. For a compre…

Isometric Immersions and Weak Solutions to the Darboux Equation
Wentao Cao, Jonas Hirsch, Dominik Inauen
https://arxiv.org/abs/2508.05230 https://arxiv.org…

Isometric Immersions and Weak Solutions to the Darboux Equation
We study the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into $\mathbb{R}^3$, in the low-regularity regime. We introduce a notion of weak solution for $u\in C^{1,θ}$ with $θ>1/2$, and show that the classical correspondence between solutions of the Darboux equation and isometric immersions remains valid in this regime. The key ingredient is an extension of the classical flatness criterion to Hölder continuous metric…

Nonlinear Splitting for Gradient-Based Unconstrained and Adjoint Optimization
Brian K. Tran, Ben S. Southworth, David B. Cavender, Sam Olivier, Syed A. Shah, Tommaso Buvoli
https://arxiv.org/abs/2508.20280

Nonlinear Splitting for Gradient-Based Unconstrained and Adjoint Optimization
High dimensional and/or nonconvex optimization remains a challenging and important problem across a wide range of fields, such as machine learning, data assimilation, and partial differential equation (PDE) constrained optimization. Here we consider gradient-based methods for solving unconstrained and constrained optimization problems, and introduce the concept of nonlinear splitting to improve accuracy and efficiency. For unconstrained optimization, we consider splittings of the gradient to de…

A general perspective on CBO methods with stochastic rate of information
Stefano Almi, Alessandro Baldi, Marco Morandotti, Francesco Solombrino
https://arxiv.org/abs/2507.20029 …

A general perspective on CBO methods with stochastic rate of information
This paper studies a class of Consensus-Based Optimization (CBO) models featuring an additional stochastic rate of information, modeling the agents' knowledge of the environment and energy landscape. The well-posedness of the stochastic system is proved, together with its finite-particle approximation and the mean-field convergence to a kinetic PDE. Particles are shown to concentrate around the consensus point under mild assumptions on the initial spatial distribution and initial level of knowl…

An Investigation into the Distribution of Ratios of Particle Solver-based Likelihoods
Emil L{\o}vbak, Sebastian Krumscheid
https://arxiv.org/abs/2508.05303 https://

An Investigation into the Distribution of Ratios of Particle Solver-based Likelihoods
We investigate the use of the Metropolis-Hastings algorithm to sample posterior distribution in a Bayesian inverse problem, where the likelihood function is random. Concretely, we consider the case where one has full field observations of a PDE solution, in case a one-dimensional diffusion equation, subject to a Gaussian observation error. Assuming one uses a particle-based Monte Carlo simulation when approximating the likelihood function, one gets an approximate likelihood with additive Gaussi…

PDE methods for extracting normal vector fields and distance functions of shapes
Takahiro Hasebe, Jun Masamune, Hiroshi Teramoto, Takayuki Yamada
https://arxiv.org/abs/2506.19323 …

PDE methods for extracting normal vector fields and distance functions of shapes
Partial differential equations can be used for extracting geometric features of shapes. This article summarizes recent methods to extract the normal vector field from an elliptic equation proposed by Yamada and from the heat equation, and also a method to extract the (signed) distance function from an elliptic equation that generalizes Varadhan's in 1967.

Fast automated adjoints for spectral PDE solvers
Calum S. Skene, Keaton J. Burns
https://arxiv.org/abs/2506.14792 https://arxiv.org/p…

Fast automated adjoints for spectral PDE solvers
We present a general and automated approach for computing model gradients for PDE solvers built on sparse spectral methods, and implement this capability in the widely used open-source Dedalus framework. We apply reverse-mode automatic differentiation to symbolic graph representations of PDEs, efficiently constructing adjoint solvers that retain the speed and memory efficiency of this important class of modern numerical methods. This approach enables users to compute gradients and perform optim…

Noise-robust multi-fidelity surrogate modelling for parametric partial differential equations
Benjamin M. Kent, Lorenzo Tamellini, Matteo Giacomini, Antonio Huerta
https://arxiv.org/abs/2507.03691

Noise-robust multi-fidelity surrogate modelling for parametric partial differential equations
We address the challenge of constructing noise-robust surrogate models for quantities of interest (QoIs) arising from parametric partial differential equations (PDEs), using multi-fidelity collocation techniques; specifically, the Multi-Index Stochastic Collocation (MISC). In practical scenarios, the PDE evaluations used to build a response surface are often corrupted by numerical noise, especially for the low-fidelity models. This noise, which may originate from loose solver tolerances, coarse…

Spectral-Prior Guided Multistage Physics-Informed Neural Networks for Highly Accurate PDE Solutions
Yuzhen Li, Liang Li, St\'ephane Lanteri, Bin Li
https://arxiv.org/abs/2508.17902

Spectral-Prior Guided Multistage Physics-Informed Neural Networks for Highly Accurate PDE Solutions
Physics-Informed Neural Networks (PINNs) are becoming a popular method for solving PDEs, due to their mesh-free nature and their ability to handle high-dimensional problems where traditional numerical solvers often struggle. Despite their promise, the practical application of PINNs is still constrained by several fac- tors, a primary one being their often-limited accuracy. This paper is dedicated to enhancing the accuracy of PINNs by introducing spectral-prior guided multistage strategy. We pro…

Fisher information for solutions of the Boltzmann equation
Cyril Imbert (DMA)
https://arxiv.org/abs/2509.03045 https://arxiv.org/pdf/2509.03045

Fisher information for solutions of the Boltzmann equation
This note reviews a recent contribution about the Fisher information for the space-homogeneous Boltzmann equation by L. Silvestre, C. Villani and the author (arXiv, 2024). This classical functional from information theory is shown to be nonincreasing along the flow of the non-linear PDE for all physically relevant particle interactions. The proof consists in establishing a new functional inequality on the sphere of Log-Sobolev type. This new a priori estimate on solutions yields global-in-time …

Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo
Joakim Beck, Yang Liu, Erik von Schwerin, Ra\'ul Tempone
https://arxiv.org/abs/2508.02925 https://

Goal-Oriented Adaptive Finite Element Multilevel Quasi-{M}onte {C}arlo
The efficient approximation of quantity of interest derived from PDEs with lognormal diffusivity is a central challenge in uncertainty quantification. In this study, we propose a multilevel quasi-Monte Carlo framework to approximate deterministic, real-valued, bounded linear functionals that depend on the solution of a linear elliptic PDE with a lognormal diffusivity coefficient {parameterized by a 49-dimensional Gaussian random vector} and deterministic geometric singularities in bounded domai…

An Iterative PDE Based Illumination Restoration Scheme for Image Enhancement
Dragos-Patru Covei
https://arxiv.org/abs/2506.12560 https://

An Iterative PDE Based Illumination Restoration Scheme for Image Enhancement
We present a novel iterative scheme for restoring uneven illumination in grayscale images. Our approach solves, at each global iteration, a nonlinear elliptic equation for an auxiliary field $u$ and then updates the illumination via an explicit time-marching step driven by the logarithmic potential of $u$. We establish existence and uniqueness of the discrete subproblems, analyze convergence of the fixed point iteration, and demonstrate performance on standard test images, reporting PSNR and SS…

ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions
Wei Cai, Andrew Qing He
https://arxiv.org/abs/2509.03757

ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions
We propose ARDO method for solving PDEs and PDE-related problems with deep learning techniques. This method uses a weak adversarial formulation but transfers the random difference operator onto the test function. The main advantage of this framework is that it is fully derivative-free with respect to the solution neural network. This framework is particularly suitable for Fokker-Planck type second-order elliptic and parabolic PDEs.

Physics-informed approach for exploratory Hamilton--Jacobi--Bellman equations via policy iterations
Yeongjong Kim, Namkyeong Cho, Minseok Kim, Yeoneung Kim
https://arxiv.org/abs/2508.01720

Physics-informed approach for exploratory Hamilton--Jacobi--Bellman equations via policy iterations
We propose a mesh-free policy iteration framework based on physics-informed neural networks (PINNs) for solving entropy-regularized stochastic control problems. The method iteratively alternates between soft policy evaluation and improvement using automatic differentiation and neural approximation, without relying on spatial discretization. We present a detailed $L^2$ error analysis that decomposes the total approximation error into three sources: iteration error, policy network error, and PDE …

Relaxation enhancement by controlling incompressible fluid flows
Kai Koike, Vahagn Nersesyan, Manuel Rissel, Marius Tucsnak
https://arxiv.org/abs/2506.22233

Relaxation enhancement by controlling incompressible fluid flows
We propose a PDE-controllability based approach to the enhancement of diffusive mixing for passive scalar fields. Unlike in the existing literature, our relaxation enhancing fields are not prescribed $\textit{ab initio}$ at every time and at every point of the spatial domain. Instead, we prove that time-dependent relaxation enhancing vector fields can be obtained as $\textit{state trajectories of control systems described by the incompressible Euler equations}$ either driven by finite-dimension…

Replaced article(s) found for math.NA. https://arxiv.org/list/math.NA/new
[1/1]:
- A Model-Consistent Data-Driven Computational Strategy for PDE Joint Inversion Problems
Kui Ren, Lu Zhang

Diff-ANO: Towards Fast High-Resolution Ultrasound Computed Tomography via Conditional Consistency Models and Adjoint Neural Operators
Xiang Cao, Qiaoqiao Ding, Xinliang Liu, Lei Zhang, Xiaoqun Zhang
https://arxiv.org/abs/2507.16344

Diff-ANO: Towards Fast High-Resolution Ultrasound Computed Tomography via Conditional Consistency Models and Adjoint Neural Operators
Ultrasound Computed Tomography (USCT) constitutes a nonlinear inverse problem with inherent ill-posedness that can benefit from regularization through diffusion generative priors. However, traditional approaches for solving Helmholtz equation-constrained USCT face three fundamental challenges when integrating these priors: PDE-constrained gradient computation, discretization-induced approximation errors, and computational imbalance between neural networks and numerical PDE solvers. In this work…

Global convergence of adaptive least-squares finite element methods for nonlinear PDEs
Philipp Bringmann, Dirk Praetorius
https://arxiv.org/abs/2509.01531 https://

Global convergence of adaptive least-squares finite element methods for nonlinear PDEs
The Zarantonello fixed-point iteration is an established linearization scheme for quasilinear PDEs with strongly monotone and Lipschitz continuous nonlinearity. This paper presents a weighted least-squares minimization for the computation of the update of this scheme. The resulting formulation allows for a conforming finite element discretization of the primal and dual variable of the PDE with arbitrary polynomial degree. The least-squares functional provides a built-in a posteriori discretizat…

A level-set based finite difference method for the ground state Bose-Einstein condensates in smooth bounded domains
Hwi Lee, Yingjie Liu
https://arxiv.org/abs/2509.00668 https:/…

A level-set based finite difference method for the ground state Bose-Einstein condensates in smooth bounded domains
We present a level-set based finite difference method to calculate the ground states of Bose Einstein condensates in domains with curved boundaries. Our method draws on the variational and level set approaches, benefiting from both of their long-standing success. More specifically, we use the normalized gradient flow, where the spatial discretization is based on the simple Cartesian grid with fictitious values in the outer vicinity of the domains. We develop a PDE-based extension technique that…

A finite element framework for simulating residential burglary in realistic urban geometries
Baoli Hao, Kamrun Mily, Annalisa Quaini, Ming Zhong
https://arxiv.org/abs/2508.11055

A finite element framework for simulating residential burglary in realistic urban geometries
We consider a partial differential equation (PDE) model to predict residential burglary derived from a probabilistic agent-based model through a mean-field limit operation. The PDE model is a nonlinear, coupled system of two equations in two variables (attractiveness of residential sites and density of criminals), similar to the Keller-Segel model for aggregation based on chemotaxis. Unlike previous works, which applied periodic boundary conditions, we enforce boundary conditions that arise nat…

Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems
Christian Engwer, Mario Ohlberger, Lukas Renelt
https://arxiv.org/abs/2507.00678

Sectional Kolmogorov N-widths for parameter-dependent function spaces: A general framework with application to parametrized Friedrichs' systems
We investigate parametrized variational problems where for each parameter the solution may originate from a different parameter-dependent function space. Our main motivation is the theory of Friedrichs' systems, a large abstract class of linear PDE-problems whose solutions are sought in operator- (and thus parameter-)dependent graph spaces. Other applications include function spaces on parametrized domains or discretizations involving data-dependent stabilizers. Concerning the set of all parame…

An energy-stable parametric finite element method for Willmore flow with normal-tangential velocity splitting
Harald Garcke, Robert N\"urnberg, Quan Zhao
https://arxiv.org/abs/2507.00193

An energy-stable parametric finite element method for Willmore flow with normal-tangential velocity splitting
We propose and analyze an energy-stable fully discrete parametric approximation for Willmore flow of hypersurfaces in two and three space dimensions. We allow for the presence of spontaneous curvature effects and for open surfaces with boundary. The presented scheme is based on a new geometric partial differential equation (PDE) that combines an evolution equation for the mean curvature with a separate equation that prescribes the tangential velocity. The mean curvature is used to determine the…

A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds
T. Chaumont-Frelet
https://arxiv.org/abs/2506.23381

A new family of a posteriori error estimates for non-conforming finite element methods leading to stabilization-free error bounds
We propose new a posteriori error estimators for non-conforming finite element discretizations of second-order elliptic PDE problems. These estimators are based on novel reformulations of the standard Prager-Synge identity, and enable to prove efficiency estimates without extra stabilization terms in the error measure for a large class of discretization schemes. We propose a residual-based estimator for which the efficiency constant scales optimally in polynomial degree, as well as two equilibr…

Preconditioned pseudo-time continuation for parameterized inverse problems
Joseph Hart, Alen Alexanderian, Bart van Bloemen Waanders
https://arxiv.org/abs/2508.21155 https://

Preconditioned pseudo-time continuation for parameterized inverse problems
We consider parametrized variational inverse problems that are constrained by partial differential equations (PDEs). We seek to efficiently compute the solution of the inverse problem when auxiliary model parameters, which appear in the governing PDE, are varied. Computing the solution of the inverse problem for different auxiliary parameter values is crucial for uncertainty quantification. This, however, is computationally challenging since it requires solving many optimization problems for di…

Random domain decomposition for parabolic PDEs on graphs
Mart\'in Hern\'andez
https://arxiv.org/abs/2508.21557 https://arxiv.org/pdf/2508.21557

Random domain decomposition for parabolic PDEs on graphs
The simulation of complex systems, such as gas transport in large pipeline networks, often involves solving PDEs posed on intricate graph structures. Such problems require considerable computational and memory resources. The Random Batch Method (RBM) has shown promise in addressing these challenges via stochastic decomposition techniques. In this paper, we apply the RBM at the PDE level for parabolic equations on graphs, without assuming any preliminary discretization in space or time. We consi…