Generalization Error Guaranteed Auto-Encoder-Based Nonlinear Model Reduction for Operator Learning
This work addresses the problem of high-dimensional data in operator learning for science and engineering, offering a method with theoretical guarantees, but it is incremental as it builds on existing auto-encoder and neural network techniques.
The paper tackles the challenge of operator learning for physical processes by proposing an Auto-Encoder-based Neural Network (AENet) for nonlinear model reduction, achieving accurate learning of solution operators for nonlinear partial differential equations with demonstrated resilience to noise.
Many physical processes in science and engineering are naturally represented by operators between infinite-dimensional function spaces. The problem of operator learning, in this context, seeks to extract these physical processes from empirical data, which is challenging due to the infinite or high dimensionality of data. An integral component in addressing this challenge is model reduction, which reduces both the data dimensionality and problem size. In this paper, we utilize low-dimensional nonlinear structures in model reduction by investigating Auto-Encoder-based Neural Network (AENet). AENet first learns the latent variables of the input data and then learns the transformation from these latent variables to corresponding output data. Our numerical experiments validate the ability of AENet to accurately learn the solution operator of nonlinear partial differential equations. Furthermore, we establish a mathematical and statistical estimation theory that analyzes the generalization error of AENet. Our theoretical framework shows that the sample complexity of training AENet is intricately tied to the intrinsic dimension of the modeled process, while also demonstrating the remarkable resilience of AENet to noise.