Physics-informed neural ODE (PINODE): embedding physics into models using collocation points

Building reduced-order models (ROMs) is essential for efficient forecasting and control of complex dynamical systems. Recently, autoencoder-based methods for building such models have gained significant traction, but their demand for data limits their use when the data is scarce and expensive. We propose aiding a model's training with the knowledge of physics using a collocation-based physics-informed loss term. Our innovation builds on ideas from classical collocation methods of numerical analysis to embed knowledge from a known equation into the latent-space dynamics of a ROM. We show that the addition of our physics-informed loss allows for exceptional data supply strategies that improves the performance of ROMs in data-scarce settings, where training high-quality data-driven models is impossible. Namely, for a problem of modeling a high-dimensional nonlinear PDE, our experiments show x 5 performance gains, measured by prediction error, in a low-data regime, x 10 performance gains in tasks of high-noise learning, x 100 gains in the efficiency of utilizing the latent-space dimension, and x 200 gains in tasks of far-out out-of-distribution forecasting relative to purely data-driven models. These improvements pave the way for broader adoption of network-based physics-informed ROMs in compressive sensing and control applications.