# Probabilistic Reduced-Order Modeling for Stochastic Partial Differential Equations

## TL;DR

This paper introduces a Bayesian, data-driven coarse-grained model for stochastic PDEs that learns microstructural features to efficiently predict fine-scale responses with quantified uncertainty.

## Contribution

It develops a physically inspired probabilistic model that captures microstructural features for efficient and uncertain-aware predictions of stochastic PDEs.

## Key findings

- Learned low-dimensional microstructural features predict fine-scale responses.
- Model provides probabilistic predictions with quantified epistemic uncertainty.
- Sparsity priors enhance feature selection and model efficiency.

## Abstract

We discuss a Bayesian formulation to coarse-graining (CG) of PDEs where the coefficients (e.g. material parameters) exhibit random, fine scale variability. The direct solution to such problems requires grids that are small enough to resolve this fine scale variability which unavoidably requires the repeated solution of very large systems of algebraic equations. We establish a physically inspired, data-driven coarse-grained model which learns a low- dimensional set of microstructural features that are predictive of the fine-grained model (FG) response. Once learned, those features provide a sharp distribution over the coarse scale effec- tive coefficients of the PDE that are most suitable for prediction of the fine scale model output. This ultimately allows to replace the computationally expensive FG by a generative proba- bilistic model based on evaluating the much cheaper CG several times. Sparsity enforcing pri- ors further increase predictive efficiency and reveal microstructural features that are important in predicting the FG response. Moreover, the model yields probabilistic rather than single-point predictions, which enables the quantification of the unavoidable epistemic uncertainty that is present due to the information loss that occurs during the coarse-graining process.