An overview on deep learning-based approximation methods for partial differential equations (Q2697278)

scientific article; zbMATH DE number 7675828

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English	An overview on deep learning-based approximation methods for partial differential equations	scientific article; zbMATH DE number 7675828

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instance of

scholarly article

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title

An overview on deep learning-based approximation methods for partial differential equations (English)

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Discrete and Continuous Dynamical Systems. Series B

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publication date

18 April 2023

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full work available at URL

https://arxiv.org/abs/2012.12348

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zbMATH Keywords

partial differential equations

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PDEs

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deep learning

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artificial neural networks

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approximation methods

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describes a project that uses

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Extensions of the deep Galerkin method

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Error analysis of the optimal quantization algorithm for obstacle problems.

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A quantization algorithm for solving multidimensional discrete-time optimal stopping problems

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Pricing Options under Rough Volatility with Backward SPDEs

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Deep Splitting Method for Parabolic PDEs

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Solving the Kolmogorov PDE by means of deep learning

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Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations

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Overcoming the curse of dimensionality in the numerical approximation of Allen-Cahn partial differential equations via truncated full-history recursive multilevel Picard approximations

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On nonlinear Feynman–Kac formulas for viscosity solutions of semilinear parabolic partial differential equations

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Numerical Simulations for Full History Recursive Multilevel Picard Approximations for Systems of High-Dimensional Partial Differential Equations

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Deep optimal stopping

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Solving high-dimensional optimal stopping problems using deep learning

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A forward scheme for backward SDEs

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Least-Squares Monte Carlo for Backward SDEs

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Time discretization and Markovian iteration for coupled FBSDEs

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Analysis of the Generalization Error: Empirical Risk Minimization over Deep Artificial Neural Networks Overcomes the Curse of Dimensionality in the Numerical Approximation of Black--Scholes Partial Differential Equations

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Solving PDEs by variational physics-informed neural networks: an a posteriori error analysis

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Variational physics informed neural networks: the role of quadratures and test functions

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Discrete-time approximation and Monte-Carlo simulation of backward stochastic differential equations

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Donsker-type theorem for BSDEs

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Simulation of BSDEs by Wiener chaos expansion

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Solving the quantum many-body problem with artificial neural networks

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Convergence Analysis of Machine Learning Algorithms for the Numerical Solution of Mean Field Control and Games I: The Ergodic Case

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Deep learning schemes for parabolic nonlocal integro-differential equations

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Machine learning for semi linear PDEs

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A branching particle system approximation for a class of FBSDEs

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Cubature method to solve BSDEs: Error expansion and complexity control

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Numerical Stability Analysis of the Euler Scheme for BSDEs

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Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions

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Extended Physics-Informed Neural Networks (XPINNs): A Generalized Space-Time Domain Decomposition Based Deep Learning Framework for Nonlinear Partial Differential Equations

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BS\(\Delta\)Es and BSDEs with non-Lipschitz drivers: comparison, convergence and robustness

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Pathwise Hölder convergence of the implicit-linear Euler scheme for semi-linear SPDEs with multiplicative noise

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Solving Backward Stochastic Differential Equations Using the Cubature Method: Application to Nonlinear Pricing

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Second order discretization of backward SDEs and simulation with the cubature method

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On the Monte Carlo simulation of BSDEs: an improvement on the Malliavin weights

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Scientific machine learning through physics-informed neural networks: where we are and what's next

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Overcoming the curse of dimensionality for some Hamilton-Jacobi partial differential equations via neural network architectures

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On some neural network architectures that can represent viscosity solutions of certain high dimensional Hamilton-Jacobi partial differential equations

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Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere

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A cubature based algorithm to solve decoupled McKean-Vlasov forward-backward stochastic differential equations

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A forward-backward stochastic algorithm for quasi-linear PDEs

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An interpolated stochastic algorithm for quasi-linear PDEs

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Neural‐network‐based approximations for solving partial differential equations

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Tensor Decomposition Methods for High-dimensional Hamilton--Jacobi--Bellman Equations

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Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations

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Algorithms for solving high dimensional PDEs: from nonlinear Monte Carlo to machine learning

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On multilevel Picard numerical approximations for high-dimensional nonlinear parabolic partial differential equations and high-dimensional nonlinear backward stochastic differential equations

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Multilevel Picard iterations for solving smooth semilinear parabolic heat equations

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The Deep Ritz Method: a deep learning-based numerical algorithm for solving variational problems

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Adaptive stochastic Galerkin FEM with hierarchical tensor representations

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DNN expression rate analysis of high-dimensional PDEs: application to option pricing

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Solving electrical impedance tomography with deep learning

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Solving inverse wave scattering with deep learning

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Multistep schemes for forward backward stochastic differential equations with jumps

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Asymptotic expansion as prior knowledge in deep learning method for high dimensional BSDEs

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Mean square rate of convergence for random walk approximation of forward-backward SDEs

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Random walk approximation of BSDEs with Hölder continuous terminal condition

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Simulation of BSDEs with jumps by Wiener chaos expansion

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Numerical solution of the parametric diffusion equation by deep neural networks

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Approximation Error Analysis of Some Deep Backward Schemes for Nonlinear PDEs

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Multilevel Monte Carlo Path Simulation

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Multilevel Monte Carlo Methods

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Solving BSDE with Adaptive Control Variate

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A regression-based Monte Carlo method to solve backward stochastic differential equations

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Stratified regression Monte-Carlo scheme for semilinear PDEs and BSDEs with large scale parallelization on GPUs

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Approximation of backward stochastic differential equations using Malliavin weights and least-squares regression

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Linear regression MDP scheme for discrete backward stochastic differential equations under general conditions

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Asset pricing with general transaction costs: Theory and numerics

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Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions

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A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations

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Deep neural network approximations for solutions of PDEs based on Monte Carlo algorithms

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Monte Carlo fPINNs: deep learning method for forward and inverse problems involving high dimensional fractional partial differential equations

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On the splitting-up method and stochastic partial differential equations

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A nonlocal physics-informed deep learning framework using the peridynamic differential operator

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Loss of regularity for Kolmogorov equations

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Convergence of deep fictitious play for stochastic differential games

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Solving high-dimensional partial differential equations using deep learning

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Convergence of the deep BSDE method for coupled FBSDEs

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Solving high-dimensional eigenvalue problems using deep neural networks: a diffusion Monte Carlo like approach

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Solving many-electron Schrödinger equation using deep neural networks

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A derivative-free method for solving elliptic partial differential equations with deep neural networks

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Monte Carlo complexity of global solution of integral equations

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Q4433869

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Monte Carlo complexity of parametric integration

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Branching diffusion representation of semilinear PDEs and Monte Carlo approximation

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A numerical algorithm for a class of BSDEs via the branching process

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Explicit Exponential Runge--Kutta Methods for Semilinear Parabolic Problems

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Exponential Runge-Kutta methods for parabolic problems.

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Malliavin calculus for backward stochastic differential equations and application to numerical solutions

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When Do Extended Physics-Informed Neural Networks (XPINNs) Improve Generalization?

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Deep backward schemes for high-dimensional nonlinear PDEs

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Overcoming the curse of dimensionality in the numerical approximation of semilinear parabolic partial differential equations

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A proof that rectified deep neural networks overcome the curse of dimensionality in the numerical approximation of semilinear heat equations

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Overcoming the curse of dimensionality in the approximative pricing of financial derivatives with default risks

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Results on Numerics for FBSDE with Drivers of Quadratic Growth

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Locally adaptive activation functions with slope recovery for deep and physics-informed neural networks

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Adaptive activation functions accelerate convergence in deep and physics-informed neural networks

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Conservative physics-informed neural networks on discrete domains for conservation laws: applications to forward and inverse problems

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Pathwise numerical approximations of SPDEs with additive noise under non-global Lipschitz coefficients

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Proof that deep artificial neural networks overcome the curse of dimensionality in the numerical approximation of Kolmogorov partial differential equations with constant diffusion and nonlinear drift coefficients

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Convergence of the deep BSDE method for FBSDEs with non-Lipschitz coefficients

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NSFnets (Navier-Stokes flow nets): physics-informed neural networks for the incompressible Navier-Stokes equations

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Simulator-free solution of high-dimensional stochastic elliptic partial differential equations using deep neural networks

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\textit{hp}-VPINNs: variational physics-informed neural networks with domain decomposition

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Solving for high-dimensional committor functions using artificial neural networks

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SwitchNet: A Neural Network Model for Forward and Inverse Scattering Problems

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Machine learning in cardiovascular flows modeling: predicting arterial blood pressure from non-invasive 4D flow MRI data using physics-informed neural networks

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A theoretical analysis of deep neural networks and parametric PDEs

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A parallel algorithm for solving BSDEs

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Particle system algorithm and chaos propagation related to non-conservative McKean type stochastic differential equations

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Monte-Carlo algorithms for a forward Feynman-Kac-type representation for semilinear nonconservative partial differential equations

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Forward Feynman-Kac type representation for semilinear non-conservative partial differential equations

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Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations

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PDE-Net 2.0: learning PDEs from data with a numeric-symbolic hybrid deep network

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Higher-Order Quasi-Monte Carlo Training of Deep Neural Networks

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DeepXDE: A Deep Learning Library for Solving Differential Equations

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Deep learning observables in computational fluid dynamics

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Iterative surrogate model optimization (ISMO): an active learning algorithm for PDE constrained optimization with deep neural networks

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Numerical method for backward stochastic differential equations

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Physics-informed neural networks for high-speed flows

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Numerical Method for Reflected Backward Stochastic Differential Equations

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PPINN: parareal physics-informed neural network for time-dependent PDEs

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Estimates on the generalization error of physics-informed neural networks for approximating a class of inverse problems for PDEs

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Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space

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Stochastic differential equations. An introduction with applications.

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nPINNs: nonlocal physics-informed neural networks for a parametrized nonlocal universal Laplacian operator. Algorithms and applications

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fPINNs: Fractional Physics-Informed Neural Networks

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Q4029028

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Adapted solution of a backward stochastic differential equation

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Probabilistic interpretation for systems of quasilinear parabolic partial differential equations

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Q3378055

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Neural networks-based backward scheme for fully nonlinear PDEs

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Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equations

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Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations

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Numerical Gaussian Processes for Time-Dependent and Nonlinear Partial Differential Equations

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Rectified deep neural networks overcome the curse of dimensionality for nonsmooth value functions in zero-sum games of nonlinear stiff systems

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Numerical simulation of BSDEs with drivers of quadratic growth

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Markovian quadratic and superquadratic BSDEs with an unbounded terminal condition

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A Fourier Cosine Method for an Efficient Computation of Solutions to BSDEs

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Numerical Fourier method and second-order Taylor scheme for backward SDEs in finance

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Unbiased Deep Solvers for Linear Parametric PDEs

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An energy approach to the solution of partial differential equations in computational mechanics via machine learning: concepts, implementation and applications

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PFNN: a penalty-free neural network method for solving a class of second-order boundary-value problems on complex geometries

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On the Convergence of Physics Informed Neural Networks for Linear Second-Order Elliptic and Parabolic Type PDEs

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DGM: a deep learning algorithm for solving partial differential equations

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Branching Diffusion Processes

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A multi-step scheme based on cubic spline for solving backward stochastic differential equations

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On optimal stopping and free boundary problems

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Deep learning of free boundary and Stefan problems

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Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks

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On the eigenvector bias of Fourier feature networks: from regression to solving multi-scale PDEs with physics-informed neural networks

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A mesh-free method for interface problems using the deep learning approach

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Nesting Monte Carlo for high-dimensional non-linear PDEs

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On the branching process for Brownian particles with an absorbing boundary

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Solving Allen-Cahn and Cahn-Hilliard Equations using the Adaptive Physics Informed Neural Networks

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Physics-Informed Generative Adversarial Networks for Stochastic Differential Equations

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Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems

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A-PINN: auxiliary physics informed neural networks for forward and inverse problems of nonlinear integro-differential equations

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Weak adversarial networks for high-dimensional partial differential equations

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Learning in Modal Space: Solving Time-Dependent Stochastic PDEs Using Physics-Informed Neural Networks

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Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems

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New Kinds of High-Order Multistep Schemes for Coupled Forward Backward Stochastic Differential Equations

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A Stable Multistep Scheme for Solving Backward Stochastic Differential Equations

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Physics-constrained deep learning for high-dimensional surrogate modeling and uncertainty quantification without labeled data

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Identifiers

Mathematics Subject Classification ID

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10.3934/DCDSB.2022238

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Sitelinks

Mathematics(1 entry)

mardi Publication:2697278