A non-parametric regularization framework for surrogate learning based on deep encoding

Many engineering applications rely on numerical methods to simulate complex physical systems, which often entail prohibitive computational costs. Additionally, problems involving parametrized domains require the generation of a computational mesh for each configuration, which is an expensive and error-prone process. Surrogates based on proper orthogonal decomposition can alleviate this burden providing rapid predictions of partial differential equations (PDEs) solutions, but they rely on explicit geometric parametrizations and shared reference meshes, requirements rarely satisfied in practice. In this paper, we present a machine-learning-based workflow for predicting PDEs solutions and derived quantities in the context of geometric parametrization, without explicit knowledge of the geometric parameters or a common discretization of the geometries. The workflow is composed of two encoding blocks that independently encode the geometries and the associated outputs, and a mapping block that learns the relationship between the two encodings. The main contribution of this work is the introduction of a regularization method for INR-based encoders, designed to preserve the structure of the encoded spaces in the learned latent space. The workflow was tested on two problems. The prediction of one-dimensional signals (radial and thrust force) on a vertical axis turbine blade with varying thickness and curvature, and the prediction of the two-dimensional pressure coefficient fields associated with geometric variations of the Royal Aircraft Establishment airfoil 2822 (RAE2822). The experimental results demonstrate that the proposed regularization substantially enhances the overall predictive accuracy, reducing the prediction error of unseen configurations by one order of magnitude compared to more standard approaches.