mlsanityData to 14 Oct 2026

Neural Q-learning for solving PDEs

arXiv 2203.17128 ↗First posted 31 Mar 2022math.NA cs.LG cs.NAcited by 8

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Abstract (arXiv)

Solving high-dimensional partial differential equations (PDEs) is a major challenge in scientific computing. We develop a new numerical method for solving elliptic-type PDEs by adapting the Q-learning algorithm in reinforcement learning. Our "Q-PDE" algorithm is mesh-free and therefore has the potential to overcome the curse of dimensionality. Using a neural tangent kernel (NTK) approach, we prove that the neural network approximator for the PDE solution, trained with the Q-PDE algorithm, converges to the trajectory of an infinite-dimensional ordinary differential equation (ODE) as the number of hidden units rightarrow ∞. For monotone PDE (i.e. those given by monotone operators, which may be nonlinear), despite the lack of a spectral gap in the NTK, we then prove that the limit neural network, which satisfies the infinite-dimensional ODE, converges in L2 to the PDE solution as the training time rightarrow ∞. More generally, we can prove that any fixed point of the wide-network limit for the Q-PDE algorithm is a solution of the PDE (not necessarily under the monotone condition). The numerical performance of the Q-PDE algorithm is studied for several elliptic PDEs.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 26 of its 57 references that are in the map. First posted on arXiv 31 Mar 2022.

Builds on

26 of 57 references.

References · 26

  1. 2014Adam: A Method for Stochastic Optimizationcited by 37,240
  2. 2013Playing Atari with Deep Reinforcement Learningcited by 3,274
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  4. 2018Neural Tangent Kernel: Convergence and Generalization in Neural Networkscited by 2,621
  5. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  6. 2019Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descentcited by 738
  7. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 678
  8. 2018On Lazy Training in Differentiable Programmingcited by 675
  9. 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
  10. 2017Deep Neural Networks as Gaussian Processescited by 622
  11. 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
  12. 2017PDE-Net: Learning PDEs from Datacited by 311
14 more
  1. 2018PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Networkcited by 210
  2. 2017Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equationscited by 191
  3. 2019Mean-field theory of two-layers neural networks: dimension-free bounds and kernel limitcited by 187
  4. 2020When Do Neural Networks Outperform Kernel Methods?cited by 150
  5. 2019Physics-Informed Neural Networks for Power Systemscited by 80
  6. 2020Algorithms for Solving High Dimensional PDEs: From Nonlinear Monte Carlo to Machine Learningcited by 77
  7. 2020An overview on deep learning-based approximation methods for partial differential equationscited by 65
  8. 2020On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEscited by 31
  9. 2021Neural networks-based algorithms for stochastic control and PDEs in financecited by 26
  10. 2020Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditionscited by 25
  11. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  12. 2019Neural networks-based backward scheme for fully nonlinear PDEscited by 23
  13. 2021PDE-constrained Models with Neural Network Terms: Optimization and Global Convergencecited by 12
  14. 2021DeepSets and their derivative networks for solving symmetric PDEscited by 10

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2023: 2; 2024: 1; 2025: 2; 2026: 3.

8 papers.

  1. 2024A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regressioncited by 7
  2. 2023The Hard-Constraint PINNs for Interface Optimal Control Problemscited by 4
  3. 2025Neural Actor-Critic Methods for Hamilton-Jacobi-Bellman PDEs: Asymptotic Analysis and Numerical Studiescited by 3
  4. 2026Deep Hilbert--Galerkin Methods for Infinite-Dimensional PDEs and Optimal Controlcited by 2
  5. 2025Random Gradient-Free Optimization in Infinite Dimensional Spacescited by 1
  6. 2026Enforcing Dirichlet Boundary Conditions in Operator Learningcited by 1
  7. 2023The ADMM-PINNs Algorithmic Framework for Nonsmooth PDE-Constrained Optimization: A Deep Learning Approachcited by 0
  8. 2026Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEscited by 0