FlowPINNs: a Variational Framework for PDE Parameter Inference and Uncertainty Quantification

Inverse problems for parameter identification in systems governed by partial differential equations (PDEs) arise in many areas of science and engineering. Traditionally, such problems have been addressed using classical numerical methods. More recently, physics-informed neural networks (PINNs) have emerged as a promising alternative for learning PDE-constrained models directly from data. However, providing principled uncertainty quantification (UQ) for the predictions obtained using PINNs remains a significant challenge. To address this limitation, we introduce flowPINNs, a probabilistic framework for estimation and UQ in PDE parameter inverse problems. The central idea is to define a variational posterior that combines a normalising flow approximation for the distribution over the PDE parameters with a parameterised PINN representing the corresponding PDE solution. This joint formulation enables efficient posterior inference via maximisation of the evidence lower bound (ELBO), thereby casting the inverse problem as a tractable optimisation task. Through a series of numerical experiments, we demonstrate that flowPINNs can achieve improved performance with strong computational efficiency when compared to existing UQ approaches for PINNs.