Pseudo-Hamiltonian neural networks for learning partial differential equations

The field called physics-informed machine learning combines the strengths of physics-based models and data-driven techniques to achieve a deeper understanding and improved predictive capabilities for complex physical systems [36, 61]. The rapidly growing interest in this interdisciplinary approach is largely motivated by the increasing capabilities of computers to store and process large quantities of data, along with the decreasing costs of sensors and computers that capture and handle data from physical systems. Machine learning for differential equations can broadly be divided into two categories: the forward problem, which involves predicting future states from an initial state, and the inverse problem, which entails learning a system or parts of it from data. A wealth of recent literature exists on machine learning for the forward problem in the context of partial differential equations (PDEs). The proposed methods include neural-network-based substitutes for numerical solvers [20, 21, 58, 53], but also methods that can aid the solution process, e.g. by optimizing the discretization to be used in a solver [2]. The focus of this paper is on the inverse problem, and much of the foundation for our proposed model can be found in recent advances in learning neural network models for ordinary differential equations (ODEs). Specifically, we build on recent works on models that incorporate Hamiltonian mechanics and related structures that underlie the physical systems we seek to model.