Tootfinder

Optimal-PhiBE: A PDE-based Model-free framework for Continuous-time Reinforcement Learning
Yuhua Zhu, Yuming Zhang, Haoyu Zhang
https://arxiv.org/abs/2506.05208

Optimal-PhiBE: A PDE-based Model-free framework for Continuous-time Reinforcement Learning
This paper addresses continuous-time reinforcement learning (CTRL) where the system dynamics are governed by a stochastic differential equation but are unknown, and only discrete-time observations are available. Existing approaches face limitations: model-based PDE methods suffer from non-identifiability, while model-free methods based on the optimal Bellman equation (Optimal-BE) are prone to large discretization errors sensitive to both the dynamics and reward structure. To overcome these chal…

This https://arxiv.org/abs/2503.17797 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLG_…

Enhancing Fourier Neural Operators with Local Spatial Features
Partial Differential Equation (PDE) problems often exhibit strong local spatial structures, and effectively capturing these structures is critical for approximating their solutions. Recently, the Fourier Neural Operator (FNO) has emerged as an efficient approach for solving these PDE problems. By using parametrization in the frequency domain, FNOs can efficiently capture global patterns. However, this approach inherently overlooks the critical role of local spatial features, as frequency-domain…

Goodness-of-fit testing for the stationary density of a size-structured PDE
Van Ha Hoang, Phu Thanh Nguyen, Thanh Mai Pham Ngoc, Vincent Rivoirard, Viet Chi Tran
https://arxiv.org/abs/2506.05103

Goodness-of-fit testing for the stationary density of a size-structured PDE
We consider two division models for structured cell populations, where cells can grow, age and divide. These models have been introduced in the literature under the denomination of `mitosis' and `adder' models. In the recent years, there has been an increasing interest in Biology to understand whether the cells divide equally or not, as this can be related to important mechanisms in cellular aging or recovery. We are therefore interested in testing the null hypothesis $H_0$ where the division o…

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…

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 …

Assessing parameter identifiability of a hemodynamics PDE model using spectral surrogates and dimension reduction
Mitchel J. Colebank
https://arxiv.org/abs/2506.04538

Assessing parameter identifiability of a hemodynamics PDE model using spectral surrogates and dimension reduction
Computational inverse problems for biomedical simulators suffer from limited data and relatively high parameter dimensionality. This often requires sensitivity analysis, where parameters of the model are ranked based on their influence on the specific quantities of interest. This is especially important for simulators used to build medical digital twins, as the amount of data is typically limited. For expensive models, such as blood flow models, emulation is employed to expedite the simulation …

This https://arxiv.org/abs/2501.16642 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

FlowDAS: A Stochastic Interpolant-based Framework for Data Assimilation
Data assimilation (DA) integrates observations with a dynamical model to estimate states of PDE-governed systems. Model-driven methods (e.g., Kalman, particle) presuppose full knowledge of the true dynamics, which is not always satisfied in practice, while purely data-driven solvers learn a deterministic mapping between observations and states and therefore miss the intrinsic stochasticity of real processes. Recently, score-based diffusion models learn a global diffusion prior and provide a goo…

This https://arxiv.org/abs/2505.10930 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csLG_…

Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models
Auto-regressive partial differential equation (PDE) foundation models have shown great potential in handling time-dependent data. However, these models suffer from the shortcut problem deeply rooted in auto-regressive prediction, causing error accumulation. The challenge becomes particularly evident for out-of-distribution data, as the pretraining performance may approach random model initialization for downstream tasks with long-term dynamics. To deal with this problem, we propose physics-info…

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…

This https://arxiv.org/abs/2308.07803 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Misspecified Bernstein-Von Mises theorem for hierarchical models
We derive a Bernstein von-Mises theorem in the context of misspecified, non-i.i.d., hierarchical models parametrized by a finite-dimensional parameter of interest. We apply our results to hierarchical models containing non-linear operators, including the squared integral operator, and PDE-constrained inverse problems. More specifically, we consider the elliptic, time-independent Schrödinger equation with parametric boundary condition and general parabolic PDEs with parametric potential and bou…

Path-dependent option pricing with two-dimensional PDE using MPDATA
Pawe{\l} Magnuszewski, Sylwester Arabas
https://arxiv.org/abs/2505.24435 https://

Path-dependent option pricing with two-dimensional PDE using MPDATA
In this paper, we discuss a simple yet robust PDE method for evaluating path-dependent Asian-style options using the non-oscillatory forward-in-time second-order MPDATA finite-difference scheme. The valuation methodology involves casting the Black-Merton-Scholes equation as a transport problem by first transforming it into a homogeneous advection-diffusion PDE via variable substitution, and then expressing the diffusion term as an advective flux using the pseudo-velocity technique. As a result,…

Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach
Shaoxuan Chen, Su Yang, Panayotis G. Kevrekidis, Wei Zhu
https://arxiv.org/abs/2506.05245

Robust Moment Identification for Nonlinear PDEs via a Neural ODE Approach
We propose a data-driven framework for learning reduced-order moment dynamics from PDE-governed systems using Neural ODEs. In contrast to derivative-based methods like SINDy, which necessitate densely sampled data and are sensitive to noise, our approach based on Neural ODEs directly models moment trajectories, enabling robust learning from sparse and potentially irregular time series. Using as an application platform the nonlinear Schrödinger equation, the framework accurately recovers govern…

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…

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.

A Budgeted Multi-Level Monte Carlo Method for Full Field Estimates of Multi-PDE Problems
Niklas Baumgarten, Robert Kutri, Robert Scheichl
https://arxiv.org/abs/2506.01644

A Budgeted Multi-Level Monte Carlo Method for Full Field Estimates of Multi-PDE Problems
We present a high-performance budgeted multi-level Monte Carlo method for estimates on the entire spatial domain of multi-PDE problems with random input data. The method is designed to operate optimally within memory and CPU-time constraints and eliminates the need for a priori knowledge of the problem's regularity and the algorithm's potential memory demand. To achieve this, we build on the budgeted multi-level Monte Carlo framework and enhance it with a sparse multi-index update algorithm ope…

This https://arxiv.org/abs/2409.15683 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qu…

Quantum DeepONet: Neural operators accelerated by quantum computing
In the realm of computational science and engineering, constructing models that reflect real-world phenomena requires solving partial differential equations (PDEs) with different conditions. Recent advancements in neural operators, such as deep operator network (DeepONet), which learn mappings between infinite-dimensional function spaces, promise efficient computation of PDE solutions for a new condition in a single forward pass. However, classical DeepONet entails quadratic complexity concerni…

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 law of large numbers for kinetic interacting diffusions
Carlo Bellingeri, Fabio Coppini
https://arxiv.org/abs/2506.01769 https://ar…

A law of large numbers for kinetic interacting diffusions
We study the convergence of the empirical distribution associated with a system of interacting kinetic particles subject to independent Brownian forcing in a finite horizon setting, using some recent progress on kinetic non-linear partial differential equations. Under general assumptions that require only weak convergence on the initial datum -without assuming independence or moment conditions- we prove convergence in probability to the corresponding non-linear Fokker-Planck PDE.

Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning
Mohammad Shah Alam, William Ott, Ilya Timofeyev
https://arxiv.org/abs/2505.24099

Attractor learning for spatiotemporally chaotic dynamical systems using echo state networks with transfer learning
In this paper, we explore the predictive capabilities of echo state networks (ESNs) for the generalized Kuramoto-Sivashinsky (gKS) equation, an archetypal nonlinear PDE that exhibits spatiotemporal chaos. We introduce a novel methodology that integrates ESNs with transfer learning, aiming to enhance predictive performance across various parameter regimes of the gKS model. Our research focuses on predicting changes in long-term statistical patterns of the gKS model that result from varying the d…

This https://arxiv.org/abs/2411.13558 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qfi…

Finding the nonnegative minimal solutions of Cauchy PDEs in a volatility-stabilized market
The strong relative arbitrage problem in Stochastic Portfolio Theory seeks an investment strategy that almost surely outperforms a benchmark portfolio at the end of a given time horizon. The highest relative return in relative arbitrage opportunities is characterized by the smallest nonnegative continuous solution of a Cauchy problem for a partial differential equation (PDE). However, solving this type of PDE poses analytical and numerical challenges, due to the high dimensionality and its non-…

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…

Deep Reinforcement Learning-Based Motion Planning and PDE Control for Flexible Manipulators
Amir Hossein Barjini, Seyed Adel Alizadeh Kolagar, Sadeq Yaqubi, Jouni Mattila
https://arxiv.org/abs/2506.08639

Deep Reinforcement Learning-Based Motion Planning and PDE Control for Flexible Manipulators
This article presents a motion planning and control framework for flexible robotic manipulators, integrating deep reinforcement learning (DRL) with a nonlinear partial differential equation (PDE) controller. Unlike conventional approaches that focus solely on control, we demonstrate that the desired trajectory significantly influences endpoint vibrations. To address this, a DRL motion planner, trained using the soft actor-critic (SAC) algorithm, generates optimized trajectories that inherently …

This https://arxiv.org/abs/2412.01625 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Aubry Set of Eikonal Hamilton-Jacobi Equations on Networks
We extend the study of eikonal Hamilton-Jacobi equations posed on networks performed by Siconolfi and Sorrentino (Anal. PDE, 2018) to a more general setting. Their approach essentially exploits that such equations correspond to discrete problems on an abstract underlying graph. However, a specific condition they assume can be rather restricting in some settings, which motivates the generalization we propose. We still get an Aubry set, which plays the role of a uniqueness set for our problem and…

This https://arxiv.org/abs/2505.18377 has been replaced.
initial toot: https://mastoxiv.page/@ar…

SP2RINT: Spatially-Decoupled Physics-Inspired Progressive Inverse Optimization for Scalable, PDE-Constrained Meta-Optical Neural Network Training
DONNs harness the physics of light propagation for efficient analog computation, with applications in AI and signal processing. Advances in nanophotonic fabrication and metasurface-based wavefront engineering have opened new pathways to realize high-capacity DONNs across various spectral regimes. Training such DONN systems to determine the metasurface structures remains challenging. Heuristic methods are fast but oversimplify metasurfaces modulation, often resulting in physically unrealizable d…

A mean field game model with non-local spatial interactions and resources accumulation
Daria Ghilli, Fausto Gozzi, Cristiano Ricci, Giovanni Zanco
https://arxiv.org/abs/2506.01200

A mean field game model with non-local spatial interactions and resources accumulation
We study a family of Mean Field Games arising in modeling the behavior of strategic economic agents which move across space maximizing their utility from consumption and have the possibility to accumulate resources for production (such as human capital). The resulting mean field game PDE system is not covered in the actual literature on the topic as it displays weaker assumptions on the regularity of the data (in particular Lipschitz continuity and boundedness of the objective are lost), state …

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

ECLEIRS: Exact conservation law embedded identification of reduced states for parameterized partial differential equations from sparse and noisy data
Aviral Prakash, Ben S. Southworth, Marc L. Klasky
https://arxiv.org/abs/2506.18855

ECLEIRS: Exact conservation law embedded identification of reduced states for parameterized partial differential equations from sparse and noisy data
Multi-query applications such as parameter estimation, uncertainty quantification and design optimization for parameterized PDE systems are expensive due to the high computational cost of high-fidelity simulations. Reduced/Latent state dynamics approaches for parameterized PDEs offer a viable method where high-fidelity data and machine learning techniques are used to reduce the system's dimensionality and estimate the dynamics of low-dimensional reduced states. These reduced state dynamics appr…

An ansatz for constructing explicit solutions of Hessian equations
Chung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang
https://arxiv.org/abs/2506.17701 https://

An ansatz for constructing explicit solutions of Hessian equations
We introduce a (variation of quadrics) ansatz for constructing explicit, real-valued solutions to broad classes of complex Hessian equations on domains in $\mathbb{C}^{n+1}$ and real Hessian equations on domains in $\mathbb{R}^{n+1}$. In the complex setting, our method simultaneously addresses the deformed Hermitian--Yang--Mills/Leung--Yau--Zaslow (dHYM/LYZ) equation, the Monge--Ampère equation, and the $J$-equation. Under this ansatz each PDE reduces to a second-order system of ordinary diffe…

Transformations of Computational Meshes
Matthew G. Knepley
https://arxiv.org/abs/2506.16341 https://arxiv.org/pdf/2506.16341

Transformations of Computational Meshes
Computational meshes, as a way to partition space, form the basis of much of PDE simulation technology, for instance for the finite element and finite volume discretization methods. In complex simulations, we are often driven to modify an input mesh, for example, to refine, coarsen, extrude, change cell types, or filter it. Mesh manipulation code can be voluminous, error-prone, spread over many special cases, and hard to understand and maintain by subsequent developers. We present a simple, tab…

An adaptive delaminating Levin method in two dimensions
Shukui Chen, Kirill Serkh, James Bremer, Murdock Aubry
https://arxiv.org/abs/2506.02424 https://

An adaptive delaminating Levin method in two dimensions
We present an adaptive delaminating Levin method for evaluating bivariate oscillatory integrals over rectangular domains. Whereas previous analyses of Levin methods impose non-resonance conditions that exclude stationary and resonance points, we rigorously establish the existence of a slowly-varying, approximate solution to the Levin PDE across all frequency regimes, even when the non-resonance condition is violated. This allows us to derive error estimates for the numerical solution of the Lev…

Mean first passage time of active Brownian particles in two dimensions
Sarafa A. Iyaniwura, Zhiwei Peng
https://arxiv.org/abs/2506.15813 https://

Mean first passage time of active Brownian particles in two dimensions
The mean first passage time (MFPT) is a key metric for understanding transport, search, and escape processes in stochastic systems. While well characterized for passive Brownian particles, its behavior in active systems-such as active Brownian particles (ABPs)-remains less understood due to their self-propelled, nonequilibrium dynamics. In this paper, we formulate and analyze an elliptic partial differential equation (PDE) to characterize the MFPT of ABPs in two-dimensional domains, including c…

Controllability and Stabilization of a Wave-Heat Cascade System
Hugo Lhachemi (L2S), Christophe Prieur (GIPSA-INFINITY), Emmanuel Tr\'elat (LJLL)
https://arxiv.org/abs/2506.10495

Controllability and Stabilization of a Wave-Heat Cascade System
Considering a wave-reaction-diffusion PDE cascade system with wave Neumann control, we first establish controllability properties in a suitable Hilbert space depending on the coupling cascade term. This is done by deriving an observability inequality for the dual problem by resorting to an Ingham-M{ü}ntz inequality. Second, we design an explicit output feedback control strategy for the actual stabilization of the PDE cascade. The key property is that the underlying operator is a Riesz-spectral…

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.

Scalable Analysis and Design Using Automatic Differentiation
Julian Andrej, Tzanio Kolev, Boyan Lazarov
https://arxiv.org/abs/2506.00746 https://

Scalable Analysis and Design Using Automatic Differentiation
This article aims to demonstrate and discuss the applications of automatic differentiation (AD) for finding derivatives in PDE-constrained optimization problems and Jacobians in non-linear finite element analysis. The main idea is to localize the application of AD at the integration point level by combining it with the so-called Finite Element Operator Decomposition. The proposed methods are computationally effective, scalable, automatic, and non-intrusive, making them ideal for existing serial…

ENMA: Tokenwise Autoregression for Generative Neural PDE Operators
Armand Kassa\"i Koupa\"i, Lise Le Boudec, Louis Serrano, Patrick Gallinari
https://arxiv.org/abs/2506.06158

ENMA: Tokenwise Autoregression for Generative Neural PDE Operators
Solving time-dependent parametric partial differential equations (PDEs) remains a fundamental challenge for neural solvers, particularly when generalizing across a wide range of physical parameters and dynamics. When data is uncertain or incomplete-as is often the case-a natural approach is to turn to generative models. We introduce ENMA, a generative neural operator designed to model spatio-temporal dynamics arising from physical phenomena. ENMA predicts future dynamics in a compressed latent …

This https://arxiv.org/abs/2309.08175 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qfi…

Closed-form solutions for VIX derivatives in a Legendre empirical model
In this paper, we introduce a data-driven, single-parameter Markov diffusion model for the VIX. The volatility factor evolves in $(-1,1)$ with a uniform invariant distribution ensured by Legendre polynomials, mapped to the empirical distribution. We derive analytical series solutions for VIX futures and options using separation of variables to solve the Feynman-Kac PDE. Compared to the 3/2 model, our approach offers equal or superior accuracy and flexibility, providing an efficient, robust alte…

Darboux formulae for linear hyperbolic equations in discrete case
Sergey V. Smirnov
https://arxiv.org/abs/2506.18603 https://arxiv.or…

Darboux formulae for linear hyperbolic equations in discrete case
In the second half of the 19th century Darboux obtained determinant formulae that provide the general solution for a linear hyperbolic second order PDE with finite Laplace series. These formulae played an important role in his study of the theory of surfaces and, in particular, in the theory of conjugate nets. During the last three decades discrete analogs of conjugate nets (Q-nets) were actively studied. Laplace series can be defined also for hyperbolic difference operators. We prove discrete …

This https://arxiv.org/abs/2412.09453 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCE_…

Finite-PINN: A Physics-Informed Neural Network with Finite Geometric Encoding for Solid Mechanics
PINN models have demonstrated capabilities in addressing fluid PDE problems, and their potential in solid mechanics is beginning to emerge. This study identifies two key challenges when using PINN to solve general solid mechanics problems. These challenges become evident when comparing the limitations of PINN with the well-established numerical methods commonly used in solid mechanics, such as the finite element method (FEM). Specifically: a) PINN models generate solutions over an infinite doma…

OmniFluids: Unified Physics Pre-trained Modeling of Fluid Dynamics
Rui Zhang, Qi Meng, Han Wan, Yang Liu, Zhi-Ming Ma, Hao Sun
https://arxiv.org/abs/2506.10862

OmniFluids: Unified Physics Pre-trained Modeling of Fluid Dynamics
High-fidelity and efficient simulation of fluid dynamics drive progress in various scientific and engineering applications. Traditional computational fluid dynamics methods offer strong interpretability and guaranteed convergence, but rely on fine spatial and temporal meshes, incurring prohibitive computational costs. Physics-informed neural networks (PINNs) and neural operators aim to accelerate PDE solvers using deep learning techniques. However, PINNs require extensive retraining and careful…

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…

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…

This https://arxiv.org/abs/2503.13193 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The deep multi-FBSDE method: a robust deep learning method for coupled FBSDEs
We introduce the deep multi-FBSDE method for robust approximation of coupled forward-backward stochastic differential equations (FBSDEs), focusing on cases where the deep BSDE method of Han, Jentzen, and E (2018) fails to converge. To overcome the convergence issues, we consider a family of FBSDEs that are equivalent to the original problem in the sense that they satisfy the same associated partial differential equation (PDE). Our algorithm proceeds in two phases: first, we approximate the init…

Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion
Darryl D. Holm, Ruiao Hu, Oliver D. Street, Hanchun Wang
https://arxiv.org/abs/2505.17026

Compound Burgers-KdV Soliton Behaviour: Refraction, Reflection and Fusion
We consider a coupled PDE system between the Burgers equation and the KdV equation to model the interactions between `bore'-like structures and wave-like solitons in shallow water. Two derivations of the resulting Burgers-swept KdV system are presented, based on Lie group symmetry and reduced variational principles. Exact compound soliton solutions are obtained, and numerical simulations show that the Burgers and KdV momenta tend toward a balance at which the coupled system reduces to the integ…

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 …

This https://arxiv.org/abs/2504.11116 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_qfi…

Breaking the Dimensional Barrier: A Pontryagin-Guided Direct Policy Optimization for Continuous-Time Multi-Asset Portfolio
Solving large-scale, continuous-time portfolio optimization problems involving numerous assets and state-dependent dynamics has long been challenged by the curse of dimensionality. Traditional dynamic programming and PDE-based methods, while rigorous, typically become computationally intractable beyond a few state variables ($\sim$3-6 limit in prior studies). To overcome this critical barrier, we introduce the \emph{Pontryagin-Guided Direct Policy Optimization} (PG-DPO) framework. PG-DPO levera…

Energy local minimizers for the nonlinear Schr\"{o}dinger equation on product spaces
Dario Pierotti, Gianmaria Verzini, Junwei Yu
https://arxiv.org/abs/2506.22371

Energy local minimizers for the nonlinear Schrödinger equation on product spaces
We investigate the existence of local minimizers with prescribed $L^2$-norm for the energy functional associated to the mass-supercritical nonlinear Schrödinger equation on the product space $\mathbb{R}^N \times M^k$, where $(M^k,g)$ is a compact Riemannian manifold, thus complementing the study of the mass-subcritical case performed by Terracini, Tzvetkov and Visciglia in [\emph{Anal. PDE} 2014, arXiv:1205.0342]. First we prove that, for small $L^2$-mass, the problem admits local minimizers…

2BSDE with uncertain horizon and application to stochastic control in erratic environments
Alberto Gennaro, Thibaut Mastrolia
https://arxiv.org/abs/2506.15037

2BSDE with uncertain horizon and application to stochastic control in erratic environments
We investigate the existence and uniqueness of non-Markovian second-order backward stochastic differential equations with an uncertain terminal horizon and establish comparison principles under the assumption that the driver is Lipschitz continuous. The terminal time is both random and exogenous, and it may not be adapted to the Brownian filtration, leading to a singular jump in the 2BSDE decomposition. We also provide a connection between this new class of 2BSDE and a fully nonlinear PDE in a …

On the Effectiveness of Classical Regression Methods for Optimal Switching Problems
Martin Andersson, Benny Avelin, Marcus Olofsson
https://arxiv.org/abs/2506.15436

On the Effectiveness of Classical Regression Methods for Optimal Switching Problems
Simple regression methods provide robust, near-optimal solutions for optimal switching problems in dimensions ranging from 1 to 50. While the theory requires solving intractable PDE systems, the Longstaff-Schwartz algorithm with classical approaches like $k$-NN achieves excellent switching decisions without extensive hyperparameter tuning. Testing eight regression approaches on four benchmark problems, we find that simple methods maintain stable performance across diverse problem characteristic…

This https://arxiv.org/abs/2312.03657 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Multisymplecticity in finite element exterior calculus
We consider the application of finite element exterior calculus (FEEC) methods to a class of canonical Hamiltonian PDE systems involving differential forms. Solutions to these systems satisfy a local multisymplectic conservation law, which generalizes the more familiar symplectic conservation law for Hamiltonian systems of ODEs, and which is connected with physically-important reciprocity phenomena, such as Lorentz reciprocity in electromagnetics. We characterize hybrid FEEC methods whose numer…

This https://arxiv.org/abs/2309.09432 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Entire solutions of two-convex Lagrangian mean curvature flows
Given an entire $C^2$ function $u$ on $\mathbb{R}^n$, we consider the graph of $D u$ as a Lagrangian submanifold of $\mathbb{R}^{2n}$, and deform it by the mean curvature flow in $\mathbb{R}^{2n}$. This leads to the special Lagrangian evolution equation, a fully nonlinear Hessian type PDE. We prove long-time existence and convergence results under a 2-positivity assumption of $(I+(D^2 u)^2)^{-1}D^2 u$. Such results were previously known only under the stronger assumption of positivity of $D^2 u…

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…

Solving partial differential equations in participating media
Bailey Miller, Rohan Sawhney, Keenan Crane, Ioannis Gkioulekas
https://arxiv.org/abs/2506.08237

Solving partial differential equations in participating media
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling this problem by treating the domain as a participating medium that models microparticle geometry stochastically, through aggregate statistical properties (e.g., particle density). We first introduce the problem setting of PDE simulation in participating medi…

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…

A New Approach for the Continuous Time Kyle-Back Strategic Insider Equilibrium Problem
Bixing Qiao, Jianfeng Zhang
https://arxiv.org/abs/2506.12281 https:/…

A New Approach for the Continuous Time Kyle-Back Strategic Insider Equilibrium Problem
This paper considers a continuous time Kyle-Back model which is a game problem between an insider and a market marker. The existing literature typically focuses on the existence of equilibrium by using the PDE approach, which requires certain Markovian structure and the equilibrium is in the bridge form. We shall provide a new approach which is used widely for stochastic controls and stochastic differential games. We characterize all equilibria through a coupled system of forward backward SDEs,…

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…

Physics-Informed Neural Networks for Control of Single-Phase Flow Systems Governed by Partial Differential Equations
Luis Kin Miyatake, Eduardo Camponogara, Eric Aislan Antonelo, Alexey Pavlov
https://arxiv.org/abs/2506.06188

Physics-Informed Neural Networks for Control of Single-Phase Flow Systems Governed by Partial Differential Equations
The modeling and control of single-phase flow systems governed by Partial Differential Equations (PDEs) present challenges, especially under transient conditions. In this work, we extend the Physics-Informed Neural Nets for Control (PINC) framework, originally proposed to modeling and control of Ordinary Differential Equations (ODE) without the need of any labeled data, to the PDE case, particularly to single-phase incompressible and compressible flows, integrating neural networks with physical…

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…

An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems
Dinshaw S. Balsara, Deepak Bhoriya, Chi-Wang Shu
https://arxiv.org/abs/2506.22312

An Alternative Finite Difference WENO-like Scheme with Physical Constraint Preservation for Divergence-Preserving Hyperbolic Systems
Alternative finite difference Weighted Essentially Non-Oscillatory (AFD-WENO) schemes allow us to very efficiently update hyperbolic systems even in complex geometries. Recent innovations in AFD-WENO methods allow us to treat hyperbolic system with non-conservative products almost as efficiently as conservation laws. However, some PDE systems,like computational electrodynamics (CED) and magnetohydrodynamics (MHD) and relativistic magnetohydrodynamics (RMHD), have involution constraints that req…

An adaptive dynamical low-rank optimizer for solving kinetic parameter identification inverse problems
Lena Baumann, Lukas Einkemmer, Christian Klingenberg, Jonas Kusch
https://arxiv.org/abs/2506.21405

An adaptive dynamical low-rank optimizer for solving kinetic parameter identification inverse problems
The numerical solution of parameter identification inverse problems for kinetic equations can exhibit high computational and memory costs. In this paper, we propose a dynamical low-rank scheme for the reconstruction of the scattering parameter in the radiative transfer equation from a number of macroscopic time-independent measurements. We first work through the PDE constrained optimization procedure in a continuous setting and derive the adjoint equations using a Lagrangian reformulation. For …

Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics
Pedro B. Bazon, Cristian G. Gebhardt, Gustavo C. Buscaglia, Roberto F. Ausas
https://arxiv.org/abs/2506.10894

Numerical approximation of a PDE-constrained Optimization problem that appears in Data-Driven Computational Mechanics
We investigate an optimization problem that arises when working within the paradigm of Data-Driven Computational Mechanics. In the context of the diffusion-reaction problem, such an optimization problem seeks for the continuous primal fields (gradient and flux) that are closest to some predefined discrete fields taken from a material data set. The optimization is performed over primal fields that satisfy the physical conservation law and the geometrical compatibility. We consider a reaction ter…

Deep random difference method for high dimensional quasilinear parabolic partial differential equations
Wei Cai, Shuixin Fang, Tao Zhou
https://arxiv.org/abs/2506.20308

Deep random difference method for high dimensional quasilinear parabolic partial differential equations
Solving high-dimensional parabolic partial differential equations (PDEs) with deep learning methods is often computationally and memory intensive, primarily due to the need for automatic differentiation (AD) to compute large Hessian matrices in the PDE. In this work, we propose a deep random difference method (DRDM) that addresses these issues by approximating the convection-diffusion operator using only first-order differences and the solution by deep neural networks, thus, avoiding explicit H…

Preconditioning and Linearly Implicit Time Integration for the Serre-Green-Naghdi Equations
Linwan Feng, David Shirokoff, Wooyoung Choi
https://arxiv.org/abs/2506.16045

Preconditioning and Linearly Implicit Time Integration for the Serre-Green-Naghdi Equations
The treatment of the differential PDE constraint poses a key challenge in computing the numerical solution of the Serre-Green-Naghdi (SGN) equations. In this work, we introduce a constant coefficient preconditioner for the SGN constraint operator and prove rigorous bounds on the preconditioned conditioning number. The conditioning bounds incorporate the effects of bathymetry in two dimensions, are quasi-optimal within a class of constant coefficient operators, highlight fundamental scalings for…

This https://arxiv.org/abs/2506.01644 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Budgeted Multi-Level Monte Carlo Method for Full Field Estimates of Multi-PDE Problems
We present a high-performance budgeted multi-level Monte Carlo method for estimates on the entire spatial domain of multi-PDE problems with random input data. The method is designed to operate optimally within memory and CPU-time constraints and eliminates the need for a priori knowledge of the problem's regularity and the algorithm's potential memory demand. To achieve this, we build on the budgeted multi-level Monte Carlo framework and enhance it with a sparse multi-index update algorithm ope…

DPG loss functions for learning parameter-to-solution maps by neural networks
Pablo Cort\'es Castillo, Wolfgang Dahmen, Jay Gopalakrishnan
https://arxiv.org/abs/2506.18773

DPG loss functions for learning parameter-to-solution maps by neural networks
We develop, analyze, and experimentally explore residual-based loss functions for machine learning of parameter-to-solution maps in the context of parameter-dependent families of partial differential equations (PDEs). Our primary concern is on rigorous accuracy certification to enhance prediction capability of resulting deep neural network reduced models. This is achieved by the use of variationally correct loss functions. Through one specific example of an elliptic PDE, details for establishin…

Weak TransNet: A Petrov-Galerkin based neural network method for solving elliptic PDEs
Zhihang Xu, Min Wang, Zhu Wang
https://arxiv.org/abs/2506.14812 http…

Weak TransNet: A Petrov-Galerkin based neural network method for solving elliptic PDEs
While deep learning has achieved remarkable success in solving partial differential equations (PDEs), it still faces significant challenges, particularly when the PDE solutions have low regularity or singularities. To address these issues, we propose the Weak TransNet (WTN) method, based on a Petrov-Galerkin formulation, for solving elliptic PDEs in this work, though its framework may extend to other classes of equations. Specifically, the neural feature space defined by TransNet (Zhang et al.,…

Convergence Analysis of a Dual-Wind Discontinuous Galerkin Method for an Elliptic Optimal Control Problem with Control Constraints
Satyajith Bommana Boyana, Thomas Lewis, Sijing Liu, Yi Zhang
https://arxiv.org/abs/2506.12330

Convergence Analysis of a Dual-Wind Discontinuous Galerkin Method for an Elliptic Optimal Control Problem with Control Constraints
This paper investigates a symmetric dual-wind discontinuous Galerkin (DWDG) method for solving an elliptic optimal control problem with control constraints. The governing constraint is an elliptic partial differential equation (PDE), which is discretized using the symmetric DWDG approach. We derive error estimates in the energy norm for both the state and the adjoint state, as well as in the $L^2$ norm of the control variable. Numerical experiments are provided to demonstrate the robustness and…

Generative Models for Parameter Space Reduction applied to Reduced Order Modelling
Guglielmo Padula, Gianluigi Rozza
https://arxiv.org/abs/2506.09721 https…

Generative Models for Parameter Space Reduction applied to Reduced Order Modelling
Solving and optimising Partial Differential Equations (PDEs) in geometrically parameterised domains often requires iterative methods, leading to high computational and time complexities. One potential solution is to learn a direct mapping from the parameters to the PDE solution. Two prominent methods for this are Data-driven Non-Intrusive Reduced Order Models (DROMs) and Parametrised Physics Informed Neural Networks (PPINNs). However, their accuracy tends to degrade as the number of geometric p…