Improving generalization ability of deep-learning-based ODE solvers using continuous dependence

Inspired by the well-known mathematical statements on the continuous dependence of solutions to ordinary differential equations on initial values and parameters, we make a non-trivial extension of the physics-informed neural networks by incorporating additional information on the continuous dependence of solutions (abbreviated as cd-PINN). Our cd-PINN integrates the advantages of neural operators and Meta-PINN, requiring only a few labeled data while enabling solving ordinary differential equations with respect to new initial values and parameters without fine-tuning. As demonstrated through a number of novel examples, the accuracy of cd-PINN under those untrained conditions is usually 1–3 orders of magnitude higher than PINN. Meanwhile, the GPU time cost for training is comparable. The extension to partial differential equations, like 2D wave equation, is also straightforward and promising. Therefore, we expect that our cd-PINN would be particularly useful in improving the efficiency and accuracy of deep-learning-based solvers for differential equations.