Variational Monte Carlo Approach to Partial Differential Equations with Neural Networks

Abstract

The accurate numerical solution of partial differential equations is a central task in numerical analysis allowing to model a wide range of natural phenomena by employing specialized solvers depending on the scenario of application. Here, we develop a variational approach for solving partial differential equations governing the evolution of high dimensional probability distributions. Our approach naturally works on the unbounded continuous domain and encodes the full probability density function through its variational parameters, which are adapted dynamically during the evolution to optimally reflect the dynamics of the density. For the considered benchmark cases we observe excellent agreement with numerical solutions as well as analytical solutions in regimes inaccessible to traditional computational approaches.

arXiv comment6 + 3 pages, 4 figures

References in corpus (17)

• Estimating Mutual Information
• DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
• Solving the Quantum Many-Body Problem with Artificial Neural Networks
• NICE: Non-linear Independent Components Estimation
• Fourier Neural Operator for Parametric Partial Differential Equations
• Quantum many-body dynamics in two dimensions with artificial neural networks
• Solving the Kolmogorov PDE by means of deep learning
• Message Passing Neural PDE Solvers
• Flow-based sampling for fermionic lattice field theories
• Viscometric Functions for a Dilute Solution of Polymers in a Good Solvent
• Coupling-based Invertible Neural Networks Are Universal Diffeomorphism Approximators
• Fokker-Planck equation of distributions of financial returns and power laws
• Unitary dynamics of strongly-interacting Bose gases with time-dependent variational Monte Carlo in continuous space
• Classical and quantum measurements of position
• jVMC: Versatile and performant variational Monte Carlo leveraging automated differentiation and GPU acceleration
• The Expressive Power of a Class of Normalizing Flow Models
• Dynamical behavior of a nonlocal Fokker-Planck equation for a stochastic system with tempered stable noise