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

Physicists use differential equations to describe the physical dynamical world, and the solutions of these equations constitute our understanding of the world. During the hundreds of years, scientists developed several ways to solve these equations, i.e., the analytical solutions and the numerical solutions. However, for some complex equations, there may be no analytical solutions, and the numerical solutions may encounter the curse of the extreme computational cost if the accuracy is the first consideration. Solving equations is a high-level human intelligence work and a crucial step towards general artificial intelligence (AI), where deep reinforcement learning (DRL) may contribute. This work makes the first attempt of applying (DRL) to solve nonlinear differential equations both in discretized and continuous format with the governing equations (physical laws) embedded in the DRL network, including ordinary differential equations (ODEs) and partial differential equations (PDEs). The DRL network consists of an actor that outputs solution approximations policy and a critic that outputs the critic of the actor's output solution. Deterministic policy network is employed as the actor, and governing equations are embedded in the critic. The effectiveness of the DRL solver in Schr\"odinger equation, Navier-Stocks, Van der Pol equation, Burgers' equation and the equation of motion are discussed.

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