General solutions for nonlinear differential equations: a rule-based self-learning approach using deep reinforcement learning
This provides a general method for solving complex differential equations in physics and engineering, though it appears incremental as it builds on existing DRL techniques.
The authors tackled the problem of solving nonlinear ordinary and partial differential equations by proposing a universal rule-based self-learning approach using deep reinforcement learning, achieving high accuracy across multiple equations including Schrödinger and Navier-Stokes.
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 Schrödinger, 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.