Functional-prior-based Bayesian PDE-constrained inversion using PINNs

2026-05-11 · Source: stat.ML updates on arXiv.org · Field: Technology & Digital — Artificial Intelligence & Machine Learning, Mathematics & Computational Sciences, Engineering & Applied Sciences · Depth: Expert, extended

Summary

This study introduces functional-prior-based approaches to Bayesian PDE-constrained inversion using physics-informed neural networks (fpBPINN), addressing the challenge of incorporating physically meaningful prior assumptions into Bayesian PINN-based inversion. The framework proposes two methods: Functional-Prior-Informed Bayesian PINN (FPI-BPINN), which learns a neural network weight prior consistent with a prescribed functional prior before performing Bayesian inference in weight space, and Function-Space Particle-Based Variational Inference for PINNs (fParVI-PINN), which conducts Bayesian estimation directly in function space using Particle-based Variational Inference (ParVI). The research highlights the critical role of random Fourier features (RFF) in enabling neural networks to accurately represent Gaussian functional priors and improve posterior approximation. The fpBPINN methods were validated through numerical experiments on one-dimensional seismic traveltime tomography and two-dimensional Darcy-flow permeability inversion, demonstrating accurate posterior distribution estimation and the significance of interpretable functional priors.

Key takeaway

For AI Scientists and Machine Learning Engineers working on PDE-constrained inverse problems, adopting functional-prior-based Bayesian PINN approaches (fpBPINN) is crucial for achieving physically interpretable uncertainty quantification. You should consider FPI-BPINN for its flexibility with various Bayesian inference methods or fParVI-PINN for its superior accuracy in directly incorporating functional priors. Integrating Random Fourier Features into your neural networks will significantly improve the representation of Gaussian functional priors and the overall posterior approximation.

Key insights

Functional priors in physics-informed neural networks improve Bayesian PDE-constrained inversion and uncertainty quantification.

Principles

• Physically meaningful priors are best expressed in function space.
• Random Fourier features enhance Gaussian functional prior representation.
• Functional priors enable physically intuitive uncertainty quantification.

Method

The fpBPINN framework offers two approaches: FPI-BPINN learns weight priors from functional priors for weight-space inference, while fParVI-PINN performs direct function-space Bayesian estimation using ParVI, both leveraging Random Fourier Features.

In practice

• Use FPI-BPINN for flexibility with existing BNN inference methods.
• Employ fParVI-PINN for higher accuracy with direct functional prior incorporation.
• Integrate Random Fourier Features in NNs for Gaussian process priors.

Topics

• Bayesian PDE-constrained Inversion
• Physics-informed Neural Networks
• Functional Prior
• Uncertainty Quantification
• Random Fourier Features

Best for: AI Scientist, Research Scientist, Machine Learning Engineer

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.