Efficient Uncertainty Quantification for Physics-Aware Machine Learning of Diffusion-Sorption Models

The Finite‑Volume Neural Network (FINN) merges the rigor of classical numerical discretizations with the flexibility of artificial neural networks (ANNs) to uncover unknown terms or parameters in partially unknown partial differential equations (PDEs). While this hybrid framework enhances flexibility and interpretability, the highly parameterized ANN makes uncertainty quantification (UQ) of the identified PDE components both demanding and computationally expensive, especially when conventional Bayesian approaches rely on costly Markov Chain Monte Carlo sampling. To address this, we introduce a computationally efficient, Machine Learning (ML)‑assisted inference‑with‑UQ scheme that yields confidence intervals for the PDE components learned by FINN. The procedure consists of data‑driven bootstrapping of the available observations and repeated training of FINN on each resampled set. We illustrate the method on the retardation factor of a diffusion‑sorption PDE, showing that it produces trustworthy interval estimates while markedly lowering the computational burden.