Deep learning based numerical approximation algorithms for stochastic partial differential equations

• URL: http://arxiv.org/abs/2012.01194v2
• Date: Mon, 20 Oct 2025 16:19:49 GMT
• Title: Deep learning based numerical approximation algorithms for stochastic partial differential equations
• Authors: Christian Beck, Sebastian Becker, Patrick Cheridito, Arnulf Jentzen, Ariel Neufeld,
• Abstract summary: We introduce and analyze a deep learning based approximation algorithm for SPDEs.<n>Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process.
• Score: 4.077787659104315
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this article, we introduce and analyze a deep learning based approximation algorithm for SPDEs. Our approach employs neural networks to approximate the solutions of SPDEs along given realizations of the driving noise process. If applied to a set of simulated noise trajectories, it yields empirical distributions of SPDE solutions, from which functionals like the mean and variance can be estimated. We test the performance of the method on stochastic heat equations with additive and multiplicative noise as well as stochastic Black-Scholes equations with multiplicative noise and Zakai equations from nonlinear filtering theory. In all cases, the proposed algorithm yields accurate results with short runtimes in up to 100 space dimensions.

Related papers

• Self-Supervised Coarsening of Unstructured Grid with Automatic Differentiation [55.88862563823878]
  In this work, we present an original algorithm to coarsen an unstructured grid based on the concepts of differentiable physics.<n>We demonstrate performance of the algorithm on two PDEs: a linear equation which governs slightly compressible fluid flow in porous media and the wave equation.<n>Our results show that in the considered scenarios, we reduced the number of grid points up to 10 times while preserving the modeled variable dynamics in the points of interest.
  arXiv  Detail & Related papers  (2025-07-24T11:02:13Z)
• Fast Expectation Value Calculation Speedup of Quantum Approximate Optimization Algorithm: HoLCUs QAOA [55.2480439325792]
  We present a new method for calculating expectation values of operators that can be expressed as a linear combination of unitary (LCU) operators. This method is general for any quantum algorithm and is of particular interest in the acceleration of variational quantum algorithms.
  arXiv  Detail & Related papers  (2025-03-03T17:15:23Z)
• Solving stochastic partial differential equations using neural networks in the Wiener chaos expansion [3.3379026542599934]
  We numerically solve partial differential equations (SPDEs) numerically by using (possibly random) neural networks in the truncated Wiener chaos expansion of their corresponding solution. We provide some approximation rates for learning the solution of SPDEs with additive and/or multiplicative noise.
  arXiv  Detail & Related papers  (2024-11-05T18:11:25Z)
• Solving Fractional Differential Equations on a Quantum Computer: A Variational Approach [0.1492582382799606]
  We introduce an efficient variational hybrid quantum-classical algorithm designed for solving Caputo time-fractional partial differential equations. Our results indicate that solution fidelity is insensitive to the fractional index and that gradient evaluation cost scales economically with the number of time steps.
  arXiv  Detail & Related papers  (2024-06-13T02:27:16Z)
• 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)
• Momentum Particle Maximum Likelihood [2.4561590439700076]
  We propose an analogous dynamical-systems-inspired approach to minimizing the free energy functional. By discretizing the system, we obtain a practical algorithm for Maximum likelihood estimation in latent variable models. The algorithm outperforms existing particle methods in numerical experiments and compares favourably with other MLE algorithms.
  arXiv  Detail & Related papers  (2023-12-12T14:53:18Z)
• Solving Systems of Linear Equations: HHL from a Tensor Networks Perspective [39.58317527488534]
  We present an algorithm for solving systems of linear equations based on the HHL algorithm with a novel qudits methodology. We perform a quantum-inspired version on tensor networks, taking advantage of their ability to perform non-unitary operations such as projection.
  arXiv  Detail & Related papers  (2023-09-11T08:18:41Z)
• Learning Unnormalized Statistical Models via Compositional Optimization [73.30514599338407]
  Noise-contrastive estimation(NCE) has been proposed by formulating the objective as the logistic loss of the real data and the artificial noise. In this paper, we study it a direct approach for optimizing the negative log-likelihood of unnormalized models.
  arXiv  Detail & Related papers  (2023-06-13T01:18:16Z)
• Stochastic Gradient Langevin Dynamics Based on Quantization with Increasing Resolution [0.0]
  We propose an alternative descent learning equation based on quantized optimization for non- objective functions. We demonstrate the effectiveness of the proposed on vanilla neural convolution neural(CNN) models and the architecture across various data sets.
  arXiv  Detail & Related papers  (2023-05-30T08:55:59Z)
• Efficient Quantum Algorithms for Nonlinear Stochastic Dynamical Systems [2.707154152696381]
  We propose efficient quantum algorithms for solving nonlinear differential equations (SDE) via the associated Fokker-Planck equation (FPE) We discretize the FPE in space and time using two well-known numerical schemes, namely Chang-Cooper and implicit finite difference. We then compute the solution of the resulting system of linear equations using the quantum linear systems.
  arXiv  Detail & Related papers  (2023-03-04T17:40:23Z)
• Optimal Algorithms for the Inhomogeneous Spiked Wigner Model [89.1371983413931]
  We derive an approximate message-passing algorithm (AMP) for the inhomogeneous problem. We identify in particular the existence of a statistical-to-computational gap where known algorithms require a signal-to-noise ratio bigger than the information-theoretic threshold to perform better than random.
  arXiv  Detail & Related papers  (2023-02-13T19:57:17Z)
• Fast Computation of Optimal Transport via Entropy-Regularized Extragradient Methods [75.34939761152587]
  Efficient computation of the optimal transport distance between two distributions serves as an algorithm that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $varepsilon$ additive accuracy.
  arXiv  Detail & Related papers  (2023-01-30T15:46:39Z)
• Deep learning numerical methods for high-dimensional fully nonlinear PIDEs and coupled FBSDEs with jumps [26.28912742740653]
  We propose a deep learning algorithm for solving high-dimensional parabolic integro-differential equations (PIDEs) The jump-diffusion process are derived by a Brownian motion and an independent compensated Poisson random measure. To derive the error estimates for this deep learning algorithm, the convergence of Markovian, the error bound of Euler time discretization, and the simulation error of deep learning algorithm are investigated.
  arXiv  Detail & Related papers  (2023-01-30T13:55:42Z)
• 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)
• Amortized Implicit Differentiation for Stochastic Bilevel Optimization [53.12363770169761]
  We study a class of algorithms for solving bilevel optimization problems in both deterministic and deterministic settings. We exploit a warm-start strategy to amortize the estimation of the exact gradient. By using this framework, our analysis shows these algorithms to match the computational complexity of methods that have access to an unbiased estimate of the gradient.
  arXiv  Detail & Related papers  (2021-11-29T15:10:09Z)
• A Stochastic Newton Algorithm for Distributed Convex Optimization [62.20732134991661]
  We analyze a Newton algorithm for homogeneous distributed convex optimization, where each machine can calculate gradients of the same population objective. We show that our method can reduce the number, and frequency, of required communication rounds compared to existing methods without hurting performance.
  arXiv  Detail & Related papers  (2021-10-07T17:51:10Z)
• Optimization and Noise Analysis of the Quantum Algorithm for Solving One-Dimensional Poisson Equation [17.65730040410185]
  We propose an efficient quantum algorithm for solving one-dimensional Poisson equation. We further develop this algorithm to make it closer to the real application on the noisy intermediate-scale quantum (NISQ) devices. We analyze the effect of common noise existing in the real quantum devices on our algorithm using the IBM Qiskit toolkit.
  arXiv  Detail & Related papers  (2021-08-27T09:44:41Z)
• Large-Scale Wasserstein Gradient Flows [84.73670288608025]
  We introduce a scalable scheme to approximate Wasserstein gradient flows. Our approach relies on input neural networks (ICNNs) to discretize the JKO steps. As a result, we can sample from the measure at each step of the gradient diffusion and compute its density.
  arXiv  Detail & Related papers  (2021-06-01T19:21:48Z)
• Optimal Randomized First-Order Methods for Least-Squares Problems [56.05635751529922]
  This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled Hadamard transforms. Our resulting algorithm yields the best complexity known for solving least-squares problems with no condition number dependence.
  arXiv  Detail & Related papers  (2020-02-21T17:45:32Z)

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.