A Mini-Batch Method for Solving Nonlinear PDEs with Gaussian Processes (2306.00307v3)

Abstract: Gaussian processes (GPs) based methods for solving partial differential equations (PDEs) demonstrate great promise by bridging the gap between the theoretical rigor of traditional numerical algorithms and the flexible design of machine learning solvers. The main bottleneck of GP methods lies in the inversion of a covariance matrix, whose cost grows cubically concerning the size of samples. Drawing inspiration from neural networks, we propose a mini-batch algorithm combined with GPs to solve nonlinear PDEs. A naive deployment of a stochastic gradient descent method for solving PDEs with GPs is challenging, as the objective function in the requisite minimization problem cannot be depicted as the expectation of a finite-dimensional random function. To address this issue, we employ a mini-batch method to the corresponding infinite-dimensional minimization problem over function spaces. The algorithm takes a mini-batch of samples at each step to update the GP model. Thus, the computational cost is allotted to each iteration. Using stability analysis and convexity arguments, we show that the mini-batch method steadily reduces a natural measure of errors towards zero at the rate of $O(1/K+1/M)$, where $K$ is the number of iterations and $M$ is the batch size.

• The paper introduces a mini-batch approach that transforms GP-based PDE solving into a stochastic proximal optimization problem, reducing computational complexity.
• The paper demonstrates that the convergence error decreases at a rate of O(1/K + 1/M), achieving accuracy comparable to full-sample GP methods.
• The paper shows that the method scales effectively for large, nonlinear PDE systems, opening avenues for future research in stochastic GP optimization.

An Analytical Review of the Mini-Batch Method for Solving Nonlinear PDEs with Gaussian Processes

This paper addresses a significant challenge in solving nonlinear partial differential equations (PDEs) using Gaussian Processes (GPs) by introducing a mini-batch methodology inspired by stochastic proximal algorithms. The traditional GP-based approach for PDEs suffers from a bottleneck due to the cubic complexity associated with covariance matrix inversion, making it computationally unattractive for large-scale problems. This research pivots on a stochastic optimization approach, employing mini-batches to update the GP model iteratively, thereby reducing computational burdens and potentially increasing the method's scalability.

Core Contributions and Methodology

The authors propose a mini-batch approach aimed at solving the infinite-dimensional minimization problem inherent in GP-based solutions for PDEs. The problem is reformulated into a stochastic optimization framework involving slack variables, effectively transforming it into a proximal optimization problem. This leverages the efficiency of mini-batches: at each step, the inversion of only the covariance matrix corresponding to the mini-batch is required, which significantly reduces computational costs to O(M^3), where M is the mini-batch size.

Key to this methodology is a novel representer theorem adapted for the mini-batch setting, permitting the reduction of each iteration to finite-dimensional optimization. This theorem outlines the dependence of the solution on the inversion of a single, small covariance matrix associated with the mini-batch.

Numerical and Theoretical Evaluations

One of the most numeric results in this paper is the convergence rate of the proposed method. The authors demonstrate that the error measure decreases at a rate of O(1/K + 1/M), with K being the number of iterations and M the mini-batch size. This denotes that errors contract with increments in either iterations or batch size, underscoring the method's efficacy.

Through extensive numerical experiments, notably solving nonlinear elliptic PDEs and Burgers' equation—the proposed approach is shown to achieve accuracy comparable with the complete sample GP method, while significantly improving computational feasibility. The convergence performance aligns with the theoretical findings, assuming bounded linear operators and weak convexity.

Implications and Future Work

From a practical standpoint, the presented mini-batch strategy enhances the scalability potential of GP-based solvers for PDEs. This is particularly crucial given the computational limitations of current methods when faced with high-dimensional data. Theoretically, this work sets a precedent for stochastic optimization's integration with GP frameworks, potentially paving the way for further research in incorporating such methods into complex PDE systems, with extensions to other GP regression problems like semi-supervised learning and hyperparameter tuning.

In future work, the paper suggests adopting different sampling techniques for selecting mini-batch samples, a factor shown to impact performance. Furthermore, exploring the integration of uncertainty quantification could enhance the robustness and adaptability of these methodologies to a wider class of PDEs and other domains.

This research contributes significantly to bridging the gap between the historically rigorous numerical approaches for PDEs and the flexible, scalable methods offered by machine learning paradigms, encapsulated by the mini-batch method delineated herein.