An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915682593013760 |
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| author | Alberts, Alex Bilionis, Ilias |
| author_facet | Alberts, Alex Bilionis, Ilias |
| contents | Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE-based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite-dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model-form error in applications, providing a diagnostic for model misspecification. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_00213 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation Alberts, Alex Bilionis, Ilias Machine Learning Computational Physics Many inverse problems require reconstructing physical fields from limited and noisy data while incorporating known governing equations. A growing body of work within probabilistic numerics formalizes such tasks via Bayesian inference in function spaces by assigning a physically meaningful prior to the latent field. In this work, we demonstrate that Brownian bridge Gaussian processes can be viewed as a softly-enforced physics-constrained prior for the Poisson equation. We first show equivalence between the variational problem associated with the Poisson equation and a kernel ridge regression objective. Then, through the connection between Gaussian process regression and kernel methods, we identify a Gaussian process for which the posterior mean function and the minimizer to the variational problem agree, thereby placing this PDE-based regularization within a fully Bayesian framework. This connection allows us to probe different theoretical questions, such as convergence and behavior of inverse problems. We then develop a finite-dimensional representation in function space and prove convergence of the projected prior and resulting posterior in Wasserstein distance. Finally, we connect the method to the important problem of identifying model-form error in applications, providing a diagnostic for model misspecification. |
| title | An interpretation of the Brownian bridge as a physics-informed prior for the Poisson equation |
| topic | Machine Learning Computational Physics |
| url | https://arxiv.org/abs/2503.00213 |