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

Inverse problems for parameter identification in systems governed by partial differential equations (PDEs) are fundamental across numerous domains in science and engineering. While traditionally approached using classical numerical techniques, recent advancements have highlighted the potential of physics-informed neural networks (PINNs) in this context. However, incorporating principled uncertainty quantification (UQ) into PINN frameworks remains a challenge. To address this, we introduce \textit{flowPINNs}, a probabilistic framework for UQ in PDE parameter inverse problems. The core idea of a flowPINN is to define a variational posterior that combines a normalising flow approximation to the distribution over the PDE parameters with a PINN that represents the corresponding PDE solution. This joint model enables efficient posterior inference by maximisation of the evidence lower bound (ELBO), thereby converting the inverse problem into a tractable optimisation task. Through a series of numerical experiments, we demonstrate that flowPINNs can yield improved predictive performance and more reliable uncertainty estimates compared to existing PINN-based UQ approaches.