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

arXiv: https://arxiv.org/abs/1805.07297 · HTML: https://mlsanity.com/paper/1805.07297/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 May 2018.

## Built on by (most cited first, 3 of 3)

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- [Learning optimal multigrid smoothers via neural networks](https://mlsanity.com/paper/2102.12071/) (2021)
- [Learning Relaxation for Multigrid](https://mlsanity.com/paper/2207.11255/) (2022)

## Abstract (arXiv)

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.