DGM: A deep learning algorithm for solving partial differential equations

Abstract

High-dimensional PDEs have been a longstanding computational challenge. We propose to solve high-dimensional PDEs by approximating the solution with a deep neural network which is trained to satisfy the differential operator, initial condition, and boundary conditions. Our algorithm is meshfree, which is key since meshes become infeasible in higher dimensions. Instead of forming a mesh, the neural network is trained on batches of randomly sampled time and space points. The algorithm is tested on a class of high-dimensional free boundary PDEs, which we are able to accurately solve in up to $200$ dimensions. The algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE and Burgers' equation. The deep learning algorithm approximates the general solution to the Burgers' equation for a continuum of different boundary conditions and physical conditions (which can be viewed as a high-dimensional space). We call the algorithm a "Deep Galerkin Method (DGM)" since it is similar in spirit to Galerkin methods, with the solution approximated by a neural network instead of a linear combination of basis functions. In addition, we prove a theorem regarding the approximation power of neural networks for a class of quasilinear parabolic PDEs.

Deep learning, machine learning, partial differential equations

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• Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations
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• Deep Domain Decomposition Method: Elliptic Problems
• A Natural Deep Ritz Method for Essential Boundary Value Problems
• Matched Asymptotic Expansions-Based Transferable Neural Networks for Singular Perturbation Problems
• Theory-Guided Machine Learning for Process Simulation of Advanced Composites
• Deep Residual Networks Learn the Geodesic Curve in the Wasserstein Space
• Efficient Shallow Ritz Method For 1D Diffusion Problems
• LMKL-Net: A Fast Localized Multiple Kernel Learning Solver via Deep Neural Networks
• Error Estimation and Correction from within Neural Network Differential Equation Solvers
• Adaptive feature capture method for solving partial differential equations with near singular solutions
• Neural-Network Chemical Emulator for First-Star Formation: Robust Iterative Predictions over a Wide Density Range
• Bayesian Reasoning for Physics Informed Neural Networks
• Highly efficient nuclear population transfer through physics-informed neural networks
• A physics-aware deep learning model for shear band formation around collapsing pores in shocked reactive materials
• Neural Stream Functions
• Probabilistic partition of unity networks for high-dimensional regression problems
• Committor functions via tensor networks
• An artificial neural network approximation for Cauchy inverse problems
• Density Propagation with Characteristics-based Deep Learning
• A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains
• A Continuous-Time Mirror Descent Approach to Sparse Phase Retrieval
• Approximation of Functionals by Neural Network without Curse of Dimensionality
• Physics-informed neural networks for non-Newtonian fluid thermo-mechanical problems: an application to rubber calendering process
• ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems
• Collocation approximation by deep neural ReLU networks for parametric elliptic PDEs with lognormal inputs
• Turbulence closure modeling with data-driven techniques: Investigation of generalizable deep neural networks
• Deep Orthogonal Decompositions for Convective Nowcasting
• Unsupervised Learning of Solutions to Differential Equations with Generative Adversarial Networks
• Optimization with learning-informed differential equation constraints and its applications
• An approximate solution for options market-making in high dimension
• Generalization Error Estimates of Machine Learning Methods for Solving High Dimensional Schrödinger Eigenvalue Problems
• Lie Transform--based Neural Networks for Dynamics Simulation and Learning
• Solving the Reaction-Diffusion equation based on analytical methods and deep learning algorithm; the Case study of sulfate attack to concrete
• A Neural Network with Plane Wave Activation for Helmholtz Equation
• Pricing spread option with liquidity adjustments
• Invertible Surrogate Models: Joint surrogate modelling and reconstruction of Laser-Wakefield Acceleration by invertible neural networks
• A pre-training deep learning method for simulating the large bending deformation of bilayer plates
• A deep learning algorithm for optimal investment strategies
• Deep Eikonal Solvers
• Deep Semi-Martingale Optimal Transport
• Normalization effects on shallow neural networks and related asymptotic expansions
• High Throughput Training of Deep Surrogates from Large Ensemble Runs
• Lower Bound on the Representation Complexity of Antisymmetric Tensor Product Functions
• PDGM: a Neural Network Approach to Solve Path-Dependent Partial Differential Equations
• Deep Domain Decomposition Method for Solving the Variational Inequality Problems
• Learning To Solve Differential Equations Across Initial Conditions
• A Novel Fourier Feature Network for Solving Partial Differential Equations
• Hopf-type representation formulas and efficient algorithms for certain high-dimensional optimal control problems
• Multi-patch isogeometric neural solver for partial differential equations on computer-aided design domains
• Inverse stochastic optimal controls
• Entanglement scaling and criticality of infinite-size quantum many-body systems in continuous space addressed by a tensor network approach
• A differential neural network learns stochastic differential equations and the Black-Scholes equation for pricing multi-asset options
• Neural network methods for Neumann series problems of Perron-Frobenius operators
• Exploring the Application of Visual Question Answering (VQA) for Classroom Activity Monitoring
• Curve fitting on a quantum annealer for an advanced navigation method
• Stationary Density Estimation of Itô Diffusions Using Deep Learning
• Weakly-supervised learning on Schrodinger equation
• Efficient Shallow Ritz Method For 1D Diffusion-Reaction Problems
• Strong $L^p$-error analysis of nonlinear Monte Carlo approximations for high-dimensional semilinear partial differential equations
• Towards Comparative Physical Interpretation of Spatial Variability Aware Neural Networks: A Summary of Results
• Traveling Wave Solutions of Partial Differential Equations via Neural Networks
• Lifetime Ruin under High-watermark Fees and Drift Uncertainty
• Deep learning for gradient flows using the Brezis-Ekeland principle
• XNet-Enhanced Deep BSDE Method and Numerical Analysis
• Physics-Aware Downsampling with Deep Learning for Scalable Flood Modeling
• Transforming physics-informed machine learning to convex optimization
• Algebraically-Informed Deep Networks (AIDN): A Deep Learning Approach to Represent Algebraic Structures