A PAC-Bayesian view of Generalisation for Physics-Informed Machine Learning

Physics-informed machine learning (PIML) integrates mechanistic knowledge, typically in the form of partial differential equations (PDEs), into data-driven models. Despite strong empirical performance, its statistical generalisation properties remain poorly understood, particularly in the regression setting with unbounded losses. Existing analyses rely largely on approximation or stability arguments and do not fully capture how physical structure influences generalisation from finite data. In this work, we develop a PAC-Bayesian framework for PIML that provides high-probability generalisation guarantees in the presence of potentially unbounded losses. Our analysis leverages the structure of physics-informed objectives to derive component-wise bounds whose complexity scales with the input-gradient energy of each loss, revealing a direct link between physical regularity and generalisation. We instantiate this framework under population-level regularity conditions and introduce a PAC-Bayesian calibration procedure that yields computable input-gradient complexities while explicitly controlling rare large-gradient events through a residual-tail correction. We then adopt a multi-task perspective that jointly treats data fidelity, PDE residuals, initial conditions, and boundary conditions, leading to a refined certificate with a single PAC-Bayesian complexity penalty, thereby avoiding the looseness induced by controlling each component independently. Based on the resulting certificate, we further develop a bound-aware learning procedure that jointly promotes empirical accuracy, proximity to a physics-informed prior, and low input-gradient complexity. Experiments across six PDE benchmarks demonstrate substantially tighter certificates than conservative bounded-loss and sub-Gaussian alternatives, while targeted ablations characterise the effects of posterior-training data, calibration data, gradient envelopes, and hypothesis localisation on the resulting guarantees.