Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints

Key points

• We prove that conditioning a Gaussian process prior jointly on linear equality observations and a finite-dimensional inequality event is equivalent to conditioning sequentially in terms of Borel measures on the solution space.1
• Under incomplete boundary data, a monotonicity constraint lowered the error of a two-dimensional Poisson reconstruction by $63%$.1
• The probability of the noisy virtual constraint event under the equality posterior measures agreement with the specified bounds before the update.1
• The construction turns admissibility from a post-processing step into conditioning information for probabilistic PDE solvers.1

Sources (1)

• [1]Constrained Probabilistic Numerics: Gaussian Process Solvers for Linear PDEs under Linear Inequality Constraints
  arXiv (AI, ML, NLP, CV, robotics, multi-agent) · Oct 9, 04:18 PM

  Gaussian process solvers for linear partial differential equations (PDEs) return posteriors rather than point estimates, but they condition only on equality information: the PDE residual, boundary data, and measurements.

  We prove that conditioning a Gaussian process prior jointly on linear equality observations and a finite-dimensional inequality event is equivalent to conditioning sequentially in terms of Borel measures on the solution space.

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