Efficient Adaptive Design of Experiments for Gradient-Enhanced Gaussian Process Surrogates in Inverse Problems

Core Concepts

A fully adaptive greedy approach to the computational design of experiments problem using gradient-enhanced Gaussian process regression as surrogates. The approach optimizes both the choice of evaluation points and the required simulation accuracy, for both values and gradients of the forward model, to efficiently construct accurate surrogate models for inverse problems.

Abstract

The paper presents a framework for efficiently constructing gradient-enhanced Gaussian process (GEGPR) surrogate models for inverse problems, where the forward model is only available through computationally expensive numerical simulations.
The key highlights are:

Formulation of the inverse problem and the GEGPR surrogate modeling approach, including the use of gradient information to improve accuracy.

Development of an adaptive design of experiments strategy that simultaneously optimizes the selection of evaluation points and the required simulation accuracy (for both values and gradients) to minimize the computational effort.

Derivation of an accuracy model based on the GEGPR posterior covariance to quantify the error in the identified parameters.

A work model that accounts for the computational cost of evaluating the forward model and its gradients.

A heuristic two-stage approach to solve the resulting non-convex optimization problem, first selecting promising evaluation points and then optimizing the tolerances.

Numerical experiments on an analytical example and a PDE-based scatterometry problem, demonstrating significant computational savings (up to 100x) by including gradient information compared to purely value-based surrogate models.

The proposed approach provides an efficient framework for constructing accurate surrogate models for inverse problems, leveraging gradient information when available to reduce the required computational effort.

Stats

The forward model y(p) is Lipschitz-continuously differentiable.
Numerical approximations yτ(p) and y'τ'(p) can be computed with error bounds ∥yτ(p) - y(p)∥ ≤ τ and ∥y'τ'(p) - y'(p)∥ ≤ τ'.
The work model for evaluating the forward model and its gradient is W(τ) = τ^(-2s) and W(τ') = c τ'^(-2s), respectively, where s = l/(2r) and l is the spatial dimension and r is the finite element order.

Quotes

"Generating simulated training data needed for constructing sufficiently accurate surrogate models to be used for efficient optimization or parameter identification can incur a huge computational effort in the offline phase."
"We make use of gradient enhanced GPR (GEGPR) promising higher accuracy than standard GPR."
"Due to the high information content of gradient data, the required number of evaluation points is small compared to gradient-free GPR."

Key Insights Distilled From

Adaptive Gradient Enhanced Gaussian Process Surrogates for Inverse Problems

by Phillip Seml... at arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01864.pdf

Deeper Inquiries

How can the reliability of the local error estimator be further improved to provide tighter bounds on the actual parameter reconstruction error

To improve the reliability of the local error estimator and provide tighter bounds on the actual parameter reconstruction error, several strategies can be implemented:

Refinement of Error Models: Enhancing the error model used in the estimation process can lead to more accurate results. This can involve incorporating additional factors that affect the error, such as noise in the measurements, uncertainties in the model, or variations in the simulation process.

Advanced Statistical Techniques: Utilizing advanced statistical techniques like Bayesian inference can help in refining the error estimates. By incorporating prior knowledge about the parameters and the measurement process, Bayesian methods can provide more robust and accurate error bounds.

Ensemble Methods: Employing ensemble methods, such as bootstrapping or cross-validation, can help in generating multiple estimates of the error and assessing their variability. This can lead to more reliable error estimates by capturing the uncertainty in the estimation process.

Sensitivity Analysis: Conducting sensitivity analysis on the error model parameters can help in understanding the impact of different factors on the error estimation. By identifying the most influential parameters, the error model can be fine-tuned to improve accuracy.

Iterative Refinement: Implementing an iterative refinement process where the error estimates are continuously updated based on new data and feedback can help in improving the reliability of the estimator over time.

By implementing these strategies and potentially combining them, the reliability of the local error estimator can be enhanced, leading to tighter bounds on the actual parameter reconstruction error.

What are the theoretical limits on the computational savings that can be achieved by including gradient information in the adaptive design of experiments, and how do they depend on the problem characteristics

The theoretical limits on the computational savings achieved by including gradient information in the adaptive design of experiments depend on several factors:

Problem Complexity: The complexity of the forward model and the sensitivity of the output to parameter variations play a significant role in determining the computational savings. In highly nonlinear and sensitive models, gradient information can lead to substantial savings compared to simpler and less sensitive models.

Gradient Computation Cost: The cost of computing gradients relative to the cost of evaluating the model function is crucial. If the gradient computation is significantly cheaper than the function evaluation, the savings can be substantial.

Accuracy Requirements: The desired accuracy level in the parameter reconstruction process influences the potential savings. If high accuracy is required, the inclusion of gradient information can lead to more efficient designs and greater computational savings.

Dimensionality of the Problem: The number of parameters and the dimensionality of the problem impact the computational savings. In high-dimensional spaces, gradient information can guide the search more effectively, potentially resulting in larger savings.

While there is no strict theoretical limit on the computational savings, the inclusion of gradient information can lead to significant efficiency gains in the adaptive design of experiments, especially in complex and high-dimensional problems with sensitive models.

Can the proposed framework be extended to handle more complex forward models, such as those involving time-dependent or stochastic partial differential equations

The proposed framework can be extended to handle more complex forward models, such as those involving time-dependent or stochastic partial differential equations, by incorporating the following adaptations:

Time-Dependent Models: For time-dependent models, the adaptive design process can be modified to account for the temporal evolution of the system. This may involve updating the evaluation points and tolerances dynamically based on the time steps and the changing behavior of the model over time.

Stochastic Models: In the case of stochastic models, the framework can be extended to incorporate probabilistic approaches for uncertainty quantification. This may involve using Bayesian methods to handle the stochastic nature of the model and the associated uncertainties in the parameters.

Multi-Fidelity Models: Extending the framework to handle multi-fidelity models can enable the integration of information from models of varying levels of complexity and accuracy. This can lead to more efficient designs by leveraging information from different fidelity levels.

Adaptive Sampling Strategies: Implementing adaptive sampling strategies tailored to the specific characteristics of time-dependent or stochastic models can enhance the efficiency of the design process. This may involve incorporating feedback mechanisms to adapt the sampling strategy based on the evolving behavior of the model.

By incorporating these adaptations and customizations, the framework can effectively handle more complex forward models, providing efficient and reliable solutions for a wide range of applications.