Scalable and Efficient Bayesian Optimal Experimental Design with Derivative-Informed Neural Operators

Core Concepts

The core message of this article is to develop an accurate, scalable, and efficient computational framework for Bayesian optimal experimental design (OED) problems by leveraging derivative-informed neural operators (DINOs). The proposed method addresses the key challenges in Bayesian OED, including the high computational cost of evaluating the parameter-to-observable (PtO) map and its derivative, the curse of dimensionality in the parameter and experimental design spaces, and the combinatorial optimization for sensor selection.

Abstract

The article presents a computational framework for solving Bayesian OED problems constrained by large-scale partial differential equation (PDE) models with high-dimensional uncertain parameters. The key contributions are:

Derivative-informed dimension reduction: The authors employ derivative-informed subspaces (DIS) for the input parameters and derivative-informed output subspaces (DOS) for the observables to achieve scalability with increasing parameter and observation dimensions.

Derivative-informed neural operators (DINOs): The authors construct DINO surrogates that accurately approximate not only the PtO map but also its derivative, which is essential for evaluating the optimality criteria in Bayesian OED.

Efficient computation of MAP point and eigenpairs: The authors formulate the optimization for the maximum a posteriori (MAP) point and the generalized eigenvalue problem for the low-rank approximation of the posterior covariance in the reduced subspaces, enabling efficient computations.

Modified swapping greedy algorithm: The authors propose a modification of the swapping greedy algorithm to optimize the experimental design by leveraging the efficient evaluations of the optimality criteria using the DINO surrogates.

The authors demonstrate the high accuracy, scalability, and efficiency of the proposed method through numerical examples with two-dimensional and three-dimensional PDE models, achieving over 1000x speedup compared to the high-fidelity Bayesian OED solutions.

Stats

The authors report the following key figures:

Over 1000x speedup for the three-dimensional PDE model, accounting for both offline construction and online evaluation costs, compared to the high-fidelity Bayesian OED solutions.
80x speedup for the two-dimensional PDE model, including both offline and online costs.

Quotes

"We develop an accurate, scalable, and efficient computational framework based on the derivative-informed neural operator (DINO) [29]."
"To achieve the computational scalability of DINO with increasing parameter and observation dimensions, we employ dimension reduction techniques by projecting the input and output to proper low-dimensional subspaces."
"Using the DINO evaluation of the PtO map and its projected Jacobian, we further derive the formulations to (1) compute the MAP point by solving a low-dimensional optimization problem using automatic differentiation with reliable derivatives, and (2) solve a small-scale eigenvalue problem in the input subspace for the low-rank approximation of the Hessian of the misfit."

Key Insights Distilled From

Accurate, scalable, and efficient Bayesian Optimal Experimental Design with derivative-informed neural operators

by Jinwoo Go,Pe... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2312.14810.pdf

Deeper Inquiries

How can the proposed DINO-based framework be extended to handle non-Gaussian priors or non-Gaussian observation noise distributions

The proposed DINO-based framework can be extended to handle non-Gaussian priors or non-Gaussian observation noise distributions by incorporating appropriate modifications in the training process of the neural network surrogate.
For non-Gaussian priors, the neural network can be trained using a loss function that accounts for the non-Gaussian nature of the prior distribution. This can be achieved by incorporating the Kullback–Leibler (KL) divergence between the true prior distribution and a Gaussian distribution as a regularization term in the loss function. By minimizing this KL divergence term along with the reconstruction error, the neural network can learn to approximate the non-Gaussian prior distribution effectively.
Similarly, for non-Gaussian observation noise distributions, the neural network can be trained to model the noise distribution accurately. This can be done by adjusting the loss function to penalize deviations from the true noise distribution, such as using a weighted loss function that gives more importance to regions where the noise distribution deviates significantly from Gaussian.
By adapting the training process of the neural network surrogate to handle non-Gaussian priors and observation noise distributions, the DINO-based framework can effectively address a wider range of probabilistic scenarios in Bayesian Optimal Experimental Design.

What are the potential limitations of the low-rank approximation approach, and how can it be further improved to handle more general PDE models or experimental design problems

The low-rank approximation approach, while effective in reducing the computational complexity of solving high-dimensional PDE models, has some potential limitations that can be further improved upon:

Limited Applicability: The low-rank approximation method relies on the assumption of fast spectral decay in the PtO map and the Hessian of the data misfit term. This assumption may not hold for all PDE models, especially those with complex or irregular behavior. Improvements can be made by developing adaptive methods that can handle a wider range of spectral decay patterns.

Accuracy vs. Efficiency Trade-off: The low-rank approximation sacrifices some accuracy for computational efficiency. Enhancements can be made by exploring hybrid approaches that combine low-rank approximations with high-fidelity computations in critical regions to improve accuracy where needed while maintaining efficiency.

Scalability: The scalability of the low-rank approximation method may be limited when dealing with extremely high-dimensional parameter spaces or observables. Future enhancements could focus on developing scalable algorithms that can handle larger dimensions without compromising accuracy.

By addressing these limitations and exploring innovative approaches, the low-rank approximation method can be further improved to handle more general PDE models and experimental design problems with increased accuracy and efficiency.

Can the derivative-informed dimension reduction techniques be applied to other types of inverse problems or uncertainty quantification tasks beyond Bayesian OED

The derivative-informed dimension reduction techniques can be applied to a wide range of inverse problems and uncertainty quantification tasks beyond Bayesian Optimal Experimental Design (OED). Some potential applications include:

Bayesian Inference: The techniques can be utilized in Bayesian inference problems to reduce the dimensionality of the parameter space and improve the efficiency of sampling methods like Markov Chain Monte Carlo (MCMC) or Variational Inference.

Optimization under Uncertainty: In optimization problems with uncertain parameters, derivative-informed dimension reduction can help in accelerating the optimization process by reducing the dimensionality of the search space while preserving important features of the problem.

Model Calibration: When calibrating complex models with uncertain parameters, these techniques can aid in efficiently exploring the parameter space and identifying the most influential parameters for model calibration.

By applying derivative-informed dimension reduction techniques to a diverse set of problems, researchers can enhance the computational efficiency and accuracy of various tasks involving high-dimensional parameter spaces and observables.