We show that the error achievable using physics-informed neural networks for solving systems of differential equations can be substantially reduced when these networks are trained using meta-learned optimization methods rather than to using fixed, hand-crafted optimizers as traditionally done. We choose a learnable optimization method based on a shallow multi-layer perceptron that is meta-trained for specific classes of differential equations. We illustrate meta-trained optimizers for several equations of practical relevance in mathematical physics, including the linear advection equation, Poisson's equation, the Korteweg--de Vries equation and Burgers' equation. We also illustrate that meta-learned optimizers exhibit transfer learning abilities, in that a meta-trained optimizer on one differential equation can also be successfully deployed on another differential equation.

翻译：我们证明，使用物理信息神经网络求解微分方程组时，若采用元学习优化方法而非传统的固定手工优化器进行训练，可显著降低可实现误差。我们选取基于浅层多层感知机的可学习优化方法，该方法针对特定类别的微分方程进行元训练。针对数学物理中若干具有实际意义的方程，包括线性平流方程、泊松方程、Korteweg–de Vries方程和Burgers方程，我们展示了元训练优化器的性能。我们还证明，元学习优化器具有迁移学习能力，即在某一微分方程上元训练的优化器也可成功应用于另一微分方程。