ARDO: A Weak Formulation Deep Neural Network Method for Elliptic and Parabolic PDEs Based on Random Differences of Test Functions

2025/09/03 by Cai, Wei, He, Andrew Qing
#35Q68 #65N99 #68T07 #76M99 #Artificial Intelligence (cs.AI) #FOS: Computer and information sciences #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2509.03757

Abstract

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.

Citations

• Deep random difference method for high-dimensional quasilinear parabolic partial differential equations
• Automatic Differentiation is Essential in Training Neural Networks for Solving Differential Equations
• SOC-MartNet: A Martingale Neural Network for the Hamilton-Jacobi-Bellman Equation without Explicit inf H in Stochastic Optimal Controls
• The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems
• DeepMartNet -- A Martingale Based Deep Neural Network Learning Method for Dirichlet BVPs and Eigenvalue Problems of Elliptic PDEs in R^d

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