General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning

TL;DR

This paper introduces a novel rule-based deep reinforcement learning method to solve nonlinear differential equations, demonstrating high accuracy and efficiency across various complex equations.

Contribution

It presents the first self-learning DRL framework for universal nonlinear differential equation solutions, leveraging transfer learning and physical rules.

Findings

Achieves high-accuracy solutions for complex equations

Demonstrates transfer learning in solving sequential tasks

Speeds up the solution process compared to traditional methods

Abstract

A universal rule-based self-learning approach using deep reinforcement learning (DRL) is proposed for the first time to solve nonlinear ordinary differential equations and partial differential equations. The solver consists of a deep neural network-structured actor that outputs candidate solutions, and a critic derived only from physical rules (governing equations and boundary and initial conditions). Solutions in discretized time are treated as multiple tasks sharing the same governing equation, and the current step parameters provide an ideal initialization for the next owing to the temporal continuity of the solutions, which shows a transfer learning characteristic and indicates that the DRL solver has captured the intrinsic nature of the equation. The approach is verified through solving the Schr\"odinger, Navier-Stokes, Burgers', Van der Pol, and Lorenz equations and an equation of motion. The results indicate that the approach gives solutions with high accuracy, and the solution process promises to get faster.