Regularized Physics-Informed Neural Networks for Parameter Estimation in Differential Equation Models

Accurate parameter estimation for ordinary differential equation (ODE) models is essential for understanding complex dynamical systems, particularly in epidemiological applications. This task is challenging in the presence of nonlinear dynamics and partial observability, where only indirect and noisy measurements of the system are available. In this article, we study parameter inference for ODE models, motivated by COVID-19 case counts in the United States during the Omicron wave. We adopt a high-dimensional compartmental model, capturing epidemiological heterogeneity where observations correspond to aggregated daily confirmed cases. We propose a regularized physics-informed neural network (PINN) framework that jointly estimates system trajectories and unknown model parameters. Unlike standard PINN approaches, which typically rely on heuristic choices of the regularization weight, we introduce a principled selection of this parameter via cross-validation to balance data fidelity and ODE constraints. This approach is conceptually related to generalized smoothing methods, but leverages neural networks for flexible trajectory representation. We establish theoretical guarantees for the proposed estimator, including convergence rates for the neural network approximation and consistency of the learned trajectories. Through simulation studies, we demonstrate that the proposed method achieves performance comparable to, and in some cases improves upon, existing approaches. The application of COVID-19 data further illustrates its effectiveness in capturing complex epidemic dynamics and providing accurate forecasts.