Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations

2025-09-05

AI Summary

Addressing the challenges of modeling solution operators and quantifying uncertainty in foundational differential equation models. Method: We propose the first operator-learning framework embedded within a stochastic differential equation (SDE) probabilistic paradigm. Our key insight is that the Contextual Operator Network (ICON) implicitly performs Bayesian inference; leveraging this, we develop a generative ICON model enabling efficient sampling from the posterior predictive distribution over solution operators—yielding principled uncertainty quantification. The method unifies Bayesian inference, SDE theory, and generative modeling to jointly handle ordinary and partial differential equations under diverse initial and boundary conditions. Contributions/Results: (i) First probabilistic generative paradigm for operator learning; (ii) theoretical identification of ICON’s implicit Bayesian nature; (iii) rigorous foundation and practical tools for multi-operator joint learning. Experiments demonstrate strong generalization and well-calibrated uncertainty estimates across heterogeneous PDE/ODE benchmarks.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Economic ramifications for generative AI infrastructure and applications

Abstract

In-context operator networks (ICON) are a class of operator learning methods based on the novel architectures of foundation models. Trained on a diverse set of datasets of initial and boundary conditions paired with corresponding solutions to ordinary and partial differential equations (ODEs and PDEs), ICON learns to map example condition-solution pairs of a given differential equation to an approximation of its solution operator. Here, we present a probabilistic framework that reveals ICON as implicitly performing Bayesian inference, where it computes the mean of the posterior predictive distribution over solution operators conditioned on the provided context, i.e., example condition-solution pairs. The formalism of random differential equations provides the probabilistic framework for describing the tasks ICON accomplishes while also providing a basis for understanding other multi-operator learning methods. This probabilistic perspective provides a basis for extending ICON to emph{generative} settings, where one can sample from the posterior predictive distribution of solution operators. The generative formulation of ICON (GenICON) captures the underlying uncertainty in the solution operator, which enables principled uncertainty quantification in the solution predictions in operator learning.

Problem

• Learning solution operators for differential equations
• Quantifying uncertainty in operator learning predictions
• Generative modeling of posterior predictive distributions

Innovation

• Probabilistic framework for Bayesian inference
• Generative sampling from posterior distribution
• Uncertainty quantification in solution predictions