Learning Generalized Hamiltonian Dynamics with Stability from Noisy Trajectory Data

We introduce a robust framework for learning various generalized Hamiltonian dynamics from noisy, sparse phase-space data and in an unsupervised manner based on variational Bayesian inference. Although conservative, dissipative, and port-Hamiltonian systems might share the same initial total energy of a closed system, it is challenging for a single Hamiltonian network model to capture the distinctive and varying motion dynamics and physics of a phase space, from sampled observational phase space trajectories. To address this complicated Hamiltonian manifold learning challenge, we extend sparse symplectic, random Fourier Gaussian processes learning with predictive successive numerical estimations of the Hamiltonian landscape, using a generalized form of state and conjugate momentum Hamiltonian dynamics, appropriate to different classes of conservative, dissipative and port-Hamiltonian physical systems. In addition to the kernelized evidence lower bound (ELBO) loss for data fidelity, we incorporate stability and conservation constraints as additional hyper-parameter balanced loss terms to regularize the model's multi-gradients, enforcing physics correctness for improved prediction accuracy with bounded uncertainty.

Robust Learning of Physics Informed Neural Networks

Physics-informed Neural Networks (PINNs) have been shown to be effective in solving partial differential equations by capturing the physics induced constraints as a part of the training loss function. This paper shows that a PINN can be sensitive to errors in training data and overfit itself in dynamically propagating these errors over the domain of the solution of the PDE. It also shows how physical regularizations based on continuity criteria and conservation laws fail to address this issue and rather introduce problems of their own causing the deep network to converge to a physics-obeying local minimum instead of the global minimum. We introduce Gaussian Process (GP) based smoothing that recovers the performance of a PINN and promises a robust architecture against noise/errors in measurements. Additionally, we illustrate an inexpensive method of quantifying the evolution of uncertainty based on the variance estimation of GPs on boundary data. Robust PINN performance is also shown to be achievable by choice of sparse sets of inducing points based on sparsely induced GPs. We demonstrate the performance of our proposed methods and compare the results from existing benchmark models in literature for time-dependent Schr\"odinger and Burgers' equations.