Experimental design is a crucial step in scientific research, as it determines the data collected and used to make conclusions. Optimal experimental design (OED) helps acquire the most informative data under constraints of budget, time, and space. This article introduces OED for Bayesian inverse problems, which involve solving an ill-posed problem using Bayesian methods. The solution requires inverting a large-scale partial differential equation (PDE), which is computationally expensive.
Keywords: Neural operators, dimension reduction, uncertainty quantification, greedy algorithm.
To tackle this challenge, the authors propose using neural networks to approximate the solution of the PDE. This allows for an offline training stage where the neural network and its Jacobian are computed, followed by an online stage where the trained model is used to solve the Bayesian OED problem without the need for large-scale PDE solves.
The authors demonstrate the efficiency of their approach through numerical experiments, showing that it can significantly reduce computational costs while maintaining accuracy. They also provide a theoretical analysis of their method’s convergence properties.
Conclusion
In summary, this article presents an efficient OED method for Bayesian inverse problems using neural operators. By offloading the computationally expensive PDE solves to an offline stage and leveraging the efficiency of neural networks, the proposed approach can significantly reduce computational costs while maintaining accuracy. The authors provide a theoretical analysis and numerical evidence supporting the effectiveness of their method. This work has important implications for researchers working in fields where Bayesian inverse problems are prevalent, such as imaging, material science, and climate modeling.
