Comparative Study of Physics-Informed Neural Networks (PINNs) and Optimizing a Discrete Loss (ODIL) for Forward and Inverse Problems in Unsteady Non-Linear Partial Differential Equations

This work systematically compares two state-of-the-art frameworks-Physics-Informed Neural Networks (PINNs) and Optimizing a Discrete Loss (ODIL)-across benchmark elliptic (Poisson), hyperbolic (Wave), and parabolic (2D diffusion-reaction with unknown source term) problems, culminating in a challenging inverse source reconstruction task: inferring a space-time-varying heat source that enforces a prescribed temperature profile along a moving line. PINNs enforce physics via continuous residuals and automatic differentiation, while ODIL discretizes the PDE on a grid and optimizes discrete field values directly. Results show ODIL consistently outperforms PINNs in convergence speed (often 10−50× faster), accuracy (lower L² errors, especially for oscillatory/high-frequency solutions), and robustness in inverse settings. Multi-scale Fourier Feature Networks (MsFFN) and Spatio-temporal Multi-scale Fourier Features Networks (STMsFFN) significantly improve PINNs performance on multi-scale problems, while ODIL with multigrid decomposition achieves comparable or superior accuracy at lower cost. The findings highlight the advantages of discrete optimization approaches over neural-network-based physics enforcement for many PDE problems, offering practical insights into hyper-parameter selection and optimization strategies. This work provides a rigorous head-to-head evaluation and guidance for choosing or combining these frameworks in computational science and engineering applications.