High-Dimensional Partial Differential Equations

In recent years, tremendous progress has been made on numerical algorithms for solving partial differential equations in a very high dimension, using ideas from either nonlinear (multilevel) Monte Carlo or deep learning. They are potentially free of the curse of dimensionality and has been proven to be so for a class of nonlinear Monte Carlo methods.

The recent review paper has reviewed these numerical and theoretical advances. In addition to algorithms based on stochastic reformulations of the original problem, such as the multilevel Picard iteration and the Deep BSDE method, we also discuss algorithms based on the more traditional Ritz, Galerkin, and least square formulations. We hope to demonstrate to the reader that studying partial differential equations as well as control and variational problems in very high dimensions might very well be among the most promising new directions in mathematics and scientific computing in the near future.

We also collect a few code repositories in this website to present the advancement of the field and promote the exploration of new problems.

An incomplete list of algorithms for high-dimensional PDEs (nonexclusive by category). Suggestions and contributions are welcome to this list.

• Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations
• Solving high-dimensional partial differential equations using deep learning
• Solving high-dimensional Hamilton-Jacobi-Bellman PDEs using neural networks: perspectives from the theory of controlled diffusions and measures on path space
• Deep splitting method for parabolic PDEs
• Numerically solving parametric families of high-dimensional Kolmogorov partial differential equations via deep learning
• Convergence of the deep BSDE method for coupled FBSDEs
• Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
• Actor-critic method for high dimensional static Hamilton–Jacobi–Bellman partial differential equations based on neural networks
• A derivative-free method for solving elliptic partial differential equations with deep neural networks
• Solving non-linear Kolmogorov equations in large dimensions by using deep learning: a numerical comparison of discretization schemes
• Deep fictitious play for finding Markovian Nash equilibrium in multi-agent games
• Optimal policies for a pandemic: A stochastic game approach and a deep learning algorithm
• Deep backward schemes for high-dimensional nonlinear PDEs
• Neural networks-based backward scheme for fully nonlinear PDEs
• A new efficient approximation scheme for solving high-dimensional semilinear PDEs: Control variate method for Deep BSDE solver
• Deep xVA solver — A neural network based counterparty credit risk management framework
• Solving high-dimensional parabolic PDEs using the tensor train format
• Deep learning algorithms for hedging with frictions
• Interpolating between BSDEs and PINNs — deep learning for elliptic and parabolic boundary value problems
• A Deep-Genetic algorithm (Deep-GA) approach for high-dimensional nonlinear parabolic partial differential equation
• Robust SDE-based variational formulations for solving linear PDEs via deep learning

Control and HJB equation:

• Adaptive deep learning for high-dimensional Hamilton-Jacobi-Bellman equations
• Approximating the stationary Bellman equation by hierarchical tensor products
• Approximative policy iteration for exit time feedback control problems driven by stochastic differential equations using tensor train format
• Tensor decomposition methods for high-dimensional Hamilton–Jacobi–Bellman equations
• Deep neural networks algorithms for stochastic control problems on finite horizon: numerical applications
• Deep learning algorithms for hedging with frictions
• A neural network approach applied to multi-agent optimal control
• A machine learning enhanced algorithm for the optimal landing problem
• Initial value problem enhanced sampling for closed-loop optimal control design with deep neural networks

Game:

• Deep fictitious play for finding Markovian Nash equilibrium in multi-agent games
• A machine learning framework for solving high-dimensional mean field game and mean field control problems
• Alternating the population and control neural networks to solve high-dimensional stochastic mean-field games
• Optimal policies for a pandemic: A stochastic game approach and a deep learning algorithm
• Random features for high-dimensional nonlocal mean-field games

Many-electron Schrödinger equation:

• Solving many-electron Schrödinger equation using deep neural networks
• Deep-neural-network solution of the electronic Schrödinger equation
• Backflow transformations via neural networks for quantum many-body wave functions
• Ab initio solution of the many-electron Schrödinger equation with deep neural networks

Commitor function:

• Solving for high dimensional committor functions using artificial neural networks
• Solving for high dimensional committor functions using neural network with online approximation to derivatives
• Active learning with importance sampling: Optimizing objectives dominated by rare events to improve generalization
• Computing committor functions for the study of rare events using deep learning
• Committor functions via tensor networks

Others:

• DGM: A deep learning algorithm for solving partial differential equations
• Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations
• An efficient data-driven solver for Fokker-Planck equations: algorithm and analysis
• A deep learning method for solving Fokker-Planck equations
• Uniformly accurate machine learning-based hydrodynamic models for kinetic equations
• Parallel tensor methods for high-dimensional linear PDEs
• Solving high-dimensional eigenvalue problems using deep neural networks: A diffusion Monte Carlo like approach
• A semigroup method for high dimensional elliptic PDEs and eigenvalue problems based on neural networks
• Tensor methods for the Boltzmann-BGK equation
• Neural parametric Fokker-Planck equation
• Neural Galerkin scheme with active learning for high-dimensional evolution equations