Error analysis for the deep Kolmogorov method

• URL: http://arxiv.org/abs/2508.17167v1
• Date: Sat, 23 Aug 2025 23:49:01 GMT
• Title: Error analysis for the deep Kolmogorov method
• Abstract summary: We provide an error analysis for the deep Kolmogorov method for heat PDEs.<n>Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN)
• Score: 6.62667679855512
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The deep Kolmogorov method is a simple and popular deep learning based method for approximating solutions of partial differential equations (PDEs) of the Kolmogorov type. In this work we provide an error analysis for the deep Kolmogorov method for heat PDEs. Specifically, we reveal convergence with convergence rates for the overall mean square distance between the exact solution of the heat PDE and the realization function of the approximating deep neural network (DNN) associated with a stochastic optimization algorithm in terms of the size of the architecture (the depth/number of hidden layers and the width of the hidden layers) of the approximating DNN, in terms of the number of random sample points used in the loss function (the number of input-output data pairs used in the loss function), and in terms of the size of the optimization error made by the employed stochastic optimization method.

Related papers

• Learning solution operators of PDEs with sparse approximation methods [2.3577368017815696]
  We investigate the approximation of solution operators for partial differential equations (PDEs) using sparse high-dimensional techniques.<n>Building on a dimension-incremental framework, we combine product basis expansions with sparse recovery methods to substantially reduce the required sample size.<n>We evaluate the resulting method numerically on several examples, comparing it against both cubature-based sparse approximation and Fourier neural operators in terms of accuracy, runtime, and sample size.
  arXiv  Detail & Related papers  (2026-06-04T11:42:09Z)
• Deep Legendre Transform [0.7734726150561086]
  We introduce a novel deep learning algorithm for computing convex conjugates of differentiable convex functions.<n>Our approach facilitates an efficient gradient-based framework for the minimization of approximation errors.<n> Numerical experiments demonstrate our method's ability to deliver accurate results across different high-dimensional examples.
  arXiv  Detail & Related papers  (2025-12-22T18:22:11Z)
• Quantitative Error Bounds for Scaling Limits of Stochastic Iterative Algorithms [10.022615790746466]
  We derive non-asymptotic functional approximation error bounds between the algorithm sample paths and the Ornstein-Uhlenbeck approximation.<n>We use our main result to construct error bounds in terms of two common metrics: the L'evy-Prokhorov and bounded Wasserstein distances.
  arXiv  Detail & Related papers  (2025-01-21T15:29:11Z)
• Total Uncertainty Quantification in Inverse PDE Solutions Obtained with Reduced-Order Deep Learning Surrogate Models [50.90868087591973]
  We propose an approximate Bayesian method for quantifying the total uncertainty in inverse PDE solutions obtained with machine learning surrogate models. We test the proposed framework by comparing it with the iterative ensemble smoother and deep ensembling methods for a non-linear diffusion equation.
  arXiv  Detail & Related papers  (2024-08-20T19:06:02Z)
• Solving Poisson Equations using Neural Walk-on-Spheres [80.1675792181381]
  We propose Neural Walk-on-Spheres (NWoS), a novel neural PDE solver for the efficient solution of high-dimensional Poisson equations. We demonstrate the superiority of NWoS in accuracy, speed, and computational costs.
  arXiv  Detail & Related papers  (2024-06-05T17:59:22Z)
• RoPINN: Region Optimized Physics-Informed Neural Networks [66.38369833561039]
  Physics-informed neural networks (PINNs) have been widely applied to solve partial differential equations (PDEs) This paper proposes and theoretically studies a new training paradigm as region optimization. A practical training algorithm, Region Optimized PINN (RoPINN), is seamlessly derived from this new paradigm.
  arXiv  Detail & Related papers  (2024-05-23T09:45:57Z)
• Differentially Private Optimization with Sparse Gradients [60.853074897282625]
  We study differentially private (DP) optimization problems under sparsity of individual gradients. Building on this, we obtain pure- and approximate-DP algorithms with almost optimal rates for convex optimization with sparse gradients.
  arXiv  Detail & Related papers  (2024-04-16T20:01:10Z)
• An Efficient Deep Learning Approach for Approximating Parameter-to-Solution Maps of PDEs [12.227294893496342]
  We propose an efficient approach combining reduced collocation methods (RCMs) and deep neural networks (DNNs)<n>In the approximation analysis section, we rigorously derive sharp upper bounds on the complexity of the neural networks.<n>The POD-DNN has demonstrated significantly accelerated computation speeds compared with conventional numerical methods.
  arXiv  Detail & Related papers  (2024-04-10T08:52:12Z)
• Multilevel CNNs for Parametric PDEs [0.0]
  We combine concepts from multilevel solvers for partial differential equations with neural network based deep learning. An in-depth theoretical analysis shows that the proposed architecture is able to approximate multigrid V-cycles to arbitrary precision. We find substantial improvements over state-of-the-art deep learning-based solvers.
  arXiv  Detail & Related papers  (2023-04-01T21:11:05Z)
• Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation [59.45669299295436]
  We propose a Monte Carlo PDE solver for training unsupervised neural solvers.<n>We use the PDEs' probabilistic representation, which regards macroscopic phenomena as ensembles of random particles.<n>Our experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency.
  arXiv  Detail & Related papers  (2023-02-10T08:05:19Z)
• An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations [68.8204255655161]
  Filtering equations play a central role in many real-life applications, including numerical weather prediction, finance and engineering. One of the classical approaches to approximate the solution of the filtering equations is to use a PDE inspired method, called the splitting-up method. We combine this method with a neural network representation to produce an approximation of the unnormalised conditional distribution of the signal process.
  arXiv  Detail & Related papers  (2022-01-10T11:01:36Z)
• A Deep Learning approach to Reduced Order Modelling of Parameter Dependent Partial Differential Equations [0.2148535041822524]
  We develop a constructive approach based on Deep Neural Networks for the efficient approximation of the parameter-to-solution map. In particular, we consider parametrized advection-diffusion PDEs, and we test the methodology in the presence of strong transport fields.
  arXiv  Detail & Related papers  (2021-03-10T17:01:42Z)
• Intermediate Layer Optimization for Inverse Problems using Deep Generative Models [86.29330440222199]
  ILO is a novel optimization algorithm for solving inverse problems with deep generative models. We empirically show that our approach outperforms state-of-the-art methods introduced in StyleGAN-2 and PULSE for a wide range of inverse problems.
  arXiv  Detail & Related papers  (2021-02-15T06:52:22Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.