Partial Differential Equations

Also known as PDE

Latest papers

51 papers12 papers in the last four weeks, up 140% on the four weeks before. 0.1% of all new papers.

Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation
Oct 22, 2026Edison Vázquez, Yingdong Lu, Tomasz NowickiNeural PDE Solvers Partial Differential Equations
Departamento de Matemáticas, CINVESTAV, México · IBM Research, Thomas J. Watson Research Center, Yorktown Heights, New York, USA
Low Residuals, Wrong Value: Learning the Viscosity Solution of Hamilton-Jacobi-Isaacs Equations with PINNs
Oct 19, 2026Raul Jimenez, Pablo Tejerina-Pérez, Leonid Sarieddine +6PDE Solving Partial Differential Equations
Institut de Ciències del Cosmos (ICCUB), Universitat de Barcelona, Barcelona, Spain · Institució Catalana de Recerca i Estudis Avançats (ICREA), Barcelona, Spain · Departament de Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain +3
GeoDiv KAN Solver for Time Fractional Navier Stokes Equations on Complex Domains to Realistic CFD Reconstruction
Oct 19, 2026Yatin Kumar, Shivani Saini, Ramesh Kumar VatsNeural PDE Solvers Partial Differential Equations
Department of Mathematics and Scientific Computing, National Institute of Technology Hamirpur, Hamirpur, Himachal Pradesh, 177005, India
Residual-Based Adaptive Refinement in Hybrid Variational-Collocation PINNs for Singular Elliptic PDEs with Mixed Dirichlet-Neumann Boundary Conditions
Oct 19, 2026Somnath Giri, Muslim Malik, Rishikesh YadavPDE Solving Partial Differential Equations
School of Mathematical & Statistical Sciences, Indian Institute of Technology Mandi, Mandi, 175005, Himachal Pradesh, India
Certified neural approximations for parabolic obstacle problems
Oct 16, 2026Davood DamircheliNeural PDE Solvers Partial Differential Equations
Department of Mathematics, Louisiana State University, Baton Rouge, 70803, LA, USA
Fractional is Better: Learnable Derivative Orders in Neural Operator Learning
Oct 14, 2026Fares B. Mehouachi, Saif Eddin JabariPDE Operator Learning Neural Operators
1New York University Abu Dhabi, Abu Dhabi, UAE · 2New York University, New York, NY, USA
Sampling from Neural Operator Priors to Solve Partial Differential Equations
Oct 12, 2026Andrew Millard, Joshua Murphy, Zheng ZhaoPDE Solving Partial Differential Equations
Department of Computer and Information Science, Linköping University, Sweden. · Department of Political Science, Trinity College Dublin, Ireland.
SBNO : Schrödinger Bridge Neural Operator for the Forward and Inverse Problems under Missing Information
Oct 9, 2026Jaehyeon Park, Inhyeok Jeong, Noseong ParkPDE Operator Learning PDE Surrogate Modeling
SolverX · KAIST
Prior or Feedback? What an LLM Uses When Adapting Neural Operators
Oct 8, 2026Julian Chan, Javier Mora JimenezPartial Differential Equations LLM Decision-Making
Compare the Market
Backward-Consistent Diffusion Sampling for Sparsely Observed PDE Inverse Problems
Oct 3, 2026Yida Pan, Muhammad H. Ashiq, Chanyong Jung +4PDE Inverse Problems Diffusion-Based Inverse Problems
University of Michigan · Los Alamos National Laboratory
Warm-starting PDE solvers with any-dimensional machine learning
Sep 30, 2026Wilson G. Gregory, George A. Kevrekidis, Ben Blum-Smith +1Neural PDE Solvers PDE Surrogate Modeling
Department of Statistics and Irving Institute for Cancer Dynamics, Columbia University · Division of Applied Mathematics, Brown University · Department of Applied Mathematics and Statistics, Johns Hopkins University
Neural Constitutive Learning for Generalized Reaction-Diffusion Systems
Sep 29, 2026Shang-Ke Chen, Yu-Peng Wang, Shih-Hsuan Hung +4Reaction-Diffusion Systems Neural PDE Solvers
National Tsing Hua University · NVIDIA
Source Anchoring for Physical Consistency in Flow Matching Models
Sep 27, 2026Giulia Romoli, Filippo Ruffini, Paolo SodaConstrained Generative Modeling Physics-Informed Generative Modeling
Department of Diagnostics and Intervention, Ume˚a University, Sweden · Unit of Artificial Intelligence and Computer Systems, Universit`a Campus Bio-Medico di Roma, Italy
Efficient Message Passing for Partial Differential Equation Priors
Sep 26, 2026Anna Kazachkova, Leonhard Hennicke, Rainer Schlosser +1PDE Solving Partial Differential Equations
Hasso Plattner Institute University of Potsdam Potsdam, Germany
Theory for groupoid equivariant neural networks: an approach for steerable CNNs on bounded domains
Sep 22, 2026Alberto Ibort, Maria Jimenez-Vazquez, Juan M. Perez-PardoEquivariant Neural Networks Partial Differential Equations
Universidad Carlos III de Madrid, ROR: 03ths8210, Departamento de Matem´aticas, Avenida de la Universidad 30 (edificio Sabatini), 28911 Legan´es (Madrid), Espa˜na. · Instituto de Ciencias Matem´aticas ICMAT (CSIC-UAM-UC3M-UCM), ROR: 05e9bn444, Campus de Cantoblanco UAM, Calle Nicol´as Cabrera 13-15, 28049 Madrid, Espa˜na.
Physics Informed Random Feature Neural Networks for Solving PDEs
Sep 14, 2026Chi-An Chen, Chunyang Liao, Ming ZhongPDE Solving Random Feature Methods
Department of Applied Mathematics, Mathematics, Illinois Institute of Technology, Chicago, IL, 60616 · Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR, 72701 · Department of Mathematics, University of Houston, Houston, TX, 77204
Selective boundary condition reduction via learned error gating
Sep 8, 2026Daniel Fernández, Dominik Penk, Dominik RiedelbauchPDE Surrogate Modeling Multifidelity Modeling
Chair for Dynamics, Control, Machine Learning, and Numerics (Alexander von Humboldt Professorship), Department of Mathematics, Friedrich-Alexander-Universität Erlangen-Nürnberg, Cauerstraße, 91058 Erlangen, Germany. · Schaeffler Technologies AG & Co. KG, Industriestraße 1–3, Herzogenaurach, Germany.
See, Hypothesize, Validate: Multimodal Agentic Framework for Discovering Governing PDEs
Aug 28, 2026Sarang Manoj Pekhale, Amartya Roy, Rajat Sarkar +1PDE Discovery Partial Differential Equations
Department of Applied Mechanics Indian Institute of Technology Delhi, India Yardi School of Artificial Intelligence (ScAI) Indian Institute of Technology Delhi, India
On Tensor-Based PDEs and their Corresponding Variational Formulations with Application to Color Image Denoising
Aug 23, 2026Freddie Åström, George Baravdish, Michael FelsbergPartial Differential Equations Image Denoising
Computer Vision Laboratory, Department of E.E., Linköping University · Center for Medical Image Science and Visualization (CMIV), Linköping University · Department of Science and Technology, Linköping University
Hypothesize, Evaluate, Refine: A Scientific Agent for PDE Discovery with Unknown Spatial Coefficient Fields
Aug 19, 2026YuJie Huang, WenWu He, ZhuoEr Lin +3PDE Discovery Partial Differential Equations
Fujian University of Technology · Fujian University of Technology Fujian Provincial Key Laboratory of Big Data Mining and Applications · Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences
MiNO: Cotangent-bundle propagator learning for PDEs
Aug 15, 2026Gnankan Landry Regis N'guessan, Bum Jun KimPDE Operator Learning Partial Differential Equations
Axiom Research Group · Department of Applied Mathematics and Computational Science, The Nelson Mandela African Institution of Science and Technology, 404 Nganana, Kikwe, Arumeru, P.O. Box 447, Arusha, 23311, Tanzania · African Institute for Mathematical Sciences, Research and Innovation Centre, KN 3 Road, Gasharu Cell, Kicukiro Sector, Kicukiro District, Kigali, Rwanda +1
Robust data-driven discovery of fractional differential equations via weak formulations and Pareto-based subset selection
Aug 13, 2026Pongpisit Thanasutives, Yoshinobu KawaharaSparse Recovery PDE Discovery
Center for Advanced Intelligence Project (AIP), RIKEN, Tokyo, Japan. · Graduate School of Information Science and Technology, The University of Osaka, Osaka, Japan.
Score-Based Stabilization for Time-Dependent Problems
Jul 27, 2026Eshed Gal, Eldad Haber, Uri AscherNeural PDE Solvers PDE Solving
University of British Columbia Vancouver, British Columbia, Canada
Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs
Jul 27, 2026Justin Sirignano, Konstantinos Spiliopoulos, Samuel CohenNeural PDE Solvers Optimization Convergence Analysis
Mathematical Institute, University of Oxford · Department of Mathematics and Statistics, Boston University
Adaptive Mamba Neural Operators
Jul 20, 2026Zeyuan Song, Zheyu JiangPDE Solving PDE Operator Learning
School of Chemical Engineering Oklahoma State University Stillwater, OK 74078, USA
SPARC-Net: A Spectral, Causality-Aware, and Hard-Constrained Physics-Informed Architecture for Stiff and Shock-Dominated Partial Differential Equations
Jul 13, 2026Divyavardhan Singh, Dimple Sonone, Hammad Mohammad +1PDE Solving Spectral Bias
Dept. of Electronics Engg. SVNIT Surat, India
Physics-Informed Neural Embeddings of PDE Solution Families
Jul 7, 2026Raul Jimenez, Svitlana Mayboroda, Pavlos Protopapas +3Intrinsic Dimensionality PDE Solving
Institute of Cosmos Sciences (ICC), University of Barcelona, Martí i Franquès 1, ES-08028, Barcelona, Spain. · Department of Mathematics, ETH Zurich, Rämistrasse 101, 8092, Zürich, Switzerland. · Institute for Applied Computational Science, Harvard University, Cambridge, MA, USA. +1
Mosaic: A Benchmark Suite for Differentiable Physics Solvers
Jun 26, 2026Andrin Rehmann, Heiko Zimmermann, Dion HäfnerAutomatic Differentiation PDE Solving
Pasteur Labs, Brooklyn, NY, USA
A Zeroth-Order Deep Learning Method for Fully Nonlinear Parabolic Partial Differential Equations with Unknown Coefficients
Jun 23, 2026Yanwei Jia, Du Ouyang, Huyên Pham +1Neural PDE Solvers PDE Solving
Department of Systems Engineering and Engineering Management The Chinese University of Hong Kong Shatin, New Territories, Hong Kong SAR · Department of Mathematical Sciences Tsinghua University Beijing, Beijing 100084, China · CMAP Ecole Polytechnique Route de Saclay, Palaiseau 91128, France +1
Agentic Symbolic Search: Characterizing PDEs Beyond Hand-crafted Expressions, Meshes, and Neural Networks
Jun 18, 2026Zongmin Yu, Liu YangAI Agents for Scientific Discovery PDE Solving
National University of Singapore

Oct 22, 2026J/K move · Enter open · S save

Convex Equilibrium Potentials for Monge--Ampère: Curvature Bounds and A Posteriori Evaluation

Edison Vázquez, Yingdong Lu, Tomasz Nowicki

Departamento de Matemáticas, CINVESTAV, México · IBM Research, Thomas J. Watson Research Center, Yorktown Heights, New York, USA

Neural PDE Solvers Partial Differential Equations Neural Network Verification Input Convex Neural Networks

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Equilibrium neural networks define their hidden state as the fixed point of a contraction, so their effective depth is set at evaluation time rather than by the architecture. Using them for fully nonlinear PDEs raises two issues that explicit networks avoid: second derivatives may grow with the effective depth, and the potential is only ever evaluated approximately. We study nonnegative softplus equilibrium networks with contraction factor

q<1

, a convex class suited to the Monge--Ampère equation. A positivity cancellation in the resolvent shows that the Hessian grows at most like

(1-q)^{-2}

, rather than the

(1-q)^{-3}

of a direct estimate, and an explicit one-neuron family shows that this exponent is sharp; third derivatives satisfy a sharp

(1-q)^{-3}

bound. Residual-based a posteriori estimates then control the state, its input derivatives and, in two dimensions, the Hessian determinant at an approximate equilibrium. Combined with interval arithmetic on complete covers of the domain, they yield rigorous uniform bounds on the Monge--Ampère residual and, through a comparison argument, on the solution error of a fixed trained network. On two manufactured Dirichlet problems on the unit disk, all six trained models are certified with residual below

0.25

and boundary error below

0.01

. Adaptive stopping halves float64 evaluation time at best, whereas certified adaptive evaluation remains costlier than fixed depth.