Variational Monte Carlo Approach to Partial Differential Equations with Neural Networks
This work addresses the problem of modeling complex natural phenomena through PDEs for researchers in numerical analysis and machine learning, though it appears incremental as it builds on existing variational and neural network methods.
The authors tackled solving high-dimensional partial differential equations for probability distributions by developing a variational Monte Carlo approach using neural networks, achieving excellent agreement with numerical and analytical solutions in regimes where traditional methods fail.
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.