BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework (2410.16420v1)

Abstract: Bayesian inference offers a robust framework for updating prior beliefs based on new data using Bayes' theorem, but exact inference is often computationally infeasible, necessitating approximate methods. Though widely used, these methods struggle to estimate marginal likelihoods accurately, particularly due to the rigid functional structures of deterministic models like Gaussian processes and the limitations of small sample sizes in stochastic models like the ensemble Kalman method. In this work, we introduce BI-EqNO, an equivariant neural operator framework for generalized approximate Bayesian inference, designed to enhance both deterministic and stochastic approaches. BI-EqNO transforms priors into posteriors conditioned on observation data through data-driven training. The framework is flexible, supporting diverse prior and posterior representations with arbitrary discretizations and varying numbers of observations. Crucially, BI-EqNO's architecture ensures (1) permutation equivariance between prior and posterior representations, and (2) permutation invariance with respect to observational data. We demonstrate BI-EqNO's utility through two examples: (1) as a generalized Gaussian process (gGP) for regression, and (2) as an ensemble neural filter (EnNF) for sequential data assimilation. Results show that gGP outperforms traditional Gaussian processes by offering a more flexible representation of covariance functions. Additionally, EnNF not only outperforms the ensemble Kalman filter in small-ensemble settings but also has the potential to function as a "super" ensemble filter, capable of representing and integrating multiple ensemble filters for enhanced assimilation performance. This study highlights BI-EqNO's versatility and effectiveness, improving Bayesian inference through data-driven training while reducing computational costs across various applications.

• The paper introduces BI-EqNO to merge neural operators with Bayesian inference, enabling the transformation of priors into posteriors while preserving key symmetries.
• It extends Gaussian processes by using trainable neural operators to model mean and covariance functions, resulting in more accurate predictions for complex functions.
• The work develops an ensemble neural filter that surpasses traditional Kalman filters by reducing assimilation errors under limited computational conditions.

Essay on "BI-EqNO: Generalized Approximate Bayesian Inference with an Equivariant Neural Operator Framework"

The paper introduces BI-EqNO, a novel framework for enhancing approximate Bayesian inference methods via an equivariant neural operator. This approach addresses limitations inherent in traditional deterministic and stochastic inference techniques, achieving improved accuracy and computational efficiency through data-driven training. The framework capitalizes on the flexibility of neural operators to transform prior distributions into posteriors by accommodating diverse representations and preserving fundamental symmetries.

Framework Overview

BI-EqNO merges concepts from neural operators with Bayesian inference, facilitating generalized transformations between prior and posterior distributions. The architecture leverages permutation equivariance and invariance, ensuring robustness across various application scenarios. This characteristic promotes adaptability for practical implementations, such as regression and sequential data assimilation, without being constrained by predefined model structures or extensive computational resources.

Generalized Gaussian Process

In the context of regression, the authors propose the generalized Gaussian process (gGP) as an extension of traditional Gaussian processes. The key innovation lies in replacing fixed kernel structures with trainable neural operators to model mean and covariance functions. This flexibility allows for more accurate predictions, particularly with complex functions exhibiting discontinuities or multiple scales. Empirical evaluations on one-dimensional discontinuous and two-dimensional multi-scale functions show that gGP outperforms classical approaches by directly inferring covariances from data rather than relying on predefined kernels.

Ensemble Neural Filter

BI-EqNO also introduces the ensemble neural filter (EnNF) for sequential data assimilation, potentially offering a more versatile substitute for ensemble Kalman filters. Training EnNF involves mimicking the performance of ensemble Kalman filters, with the results indicating superior performance, especially in scenarios constrained by limited ensemble sizes. The inherent non-linear update scheme in EnNF is capable of overcoming the limitations of linear ensemble updates, characteristic of Kalman filters, thus presenting notable advantages under restricted computational conditions.

Numerical Results

The numerical experiments conducted using the Lorenz-63 and Lorenz-96 models reinforce the potential of EnNF. These tests demonstrate that EnNF consistently achieves lower assimilation errors compared to ensemble Kalman filters, particularly under small ensemble settings. This resilience and efficiency underscore its suitability for real-world applications where computational power may be limited.

Implications and Future Work

The BI-EqNO framework provides a promising direction for advancing probabilistic inference methods. By integrating neural operators, it allows for significant improvements in modeling flexibility and computational performance. Future research would benefit from further exploration of BI-EqNO's application in real-world scenarios, including incorporating physical constraints into the gGP and enhancing the scalability of EnNF for high-dimensional systems.

In conclusion, BI-EqNO represents a significant step in the evolution of approximate Bayesian inference, offering enhanced flexibility and performance potential for a broad range of applications within AI and beyond. Its development could catalyze further innovations in leveraging neural operators for complex probabilistic modeling challenges.