# Bayesian Nonlinear PDE Inference via Gaussian Process Collocation with Application to the Richards Equation

arXiv: https://arxiv.org/abs/2510.23550 · HTML: https://mlsanity.com/paper/2510.23550/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 51 references that are in the map.
- First posted on arXiv 27 Oct 2025.

## Builds on (most cited first, 8 of 8)

- [A Tutorial on Bayesian Optimization](https://mlsanity.com/paper/1807.02811/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Dimension-independent likelihood-informed MCMC](https://mlsanity.com/paper/1411.3688/) (2014)
- [Probabilistic Numerical Methods for PDE-constrained Bayesian Inverse Problems](https://mlsanity.com/paper/1701.04006/) (2017)
- [Bayesian Numerical Methods for Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2104.12587/) (2021)
- [Improving multiple-try Metropolis with local balancing](https://mlsanity.com/paper/2211.11613/) (2022)
- [Parameter Inference based on Gaussian Processes Informed by Nonlinear Partial Differential Equations](https://mlsanity.com/paper/2212.11880/) (2022)
- [Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry](https://mlsanity.com/paper/2509.02617/) (2025)

## Abstract (arXiv)

The estimation of unknown parameters in nonlinear partial differential equations (PDEs) offers valuable insights across a wide range of scientific domains. In this work, we focus on estimating plant root parameters in the Richards equation, which is essential for understanding the soil-plant system in agricultural studies. Since conventional methods are computationally intensive and often yield unstable estimates, we develop a new Gaussian process collocation method for efficient Bayesian inference. Unlike existing Gaussian process-based approaches, our method constructs an approximate posterior distribution using samples drawn from a Gaussian process model fitted to the observed data, which does not require any structural assumption about the underlying PDE. Further, we propose to use an importance sampling procedure to correct for the discrepancy between the approximate and true posterior distributions. As an alternative, we also devise a prior-guided Bayesian optimization algorithm leveraging the approximate posterior. Simulation studies demonstrate that our method yields robust estimates under various settings. Finally, we apply our method on a real agricultural data set and estimate the plant root parameters with uncertainty quantification.