An adaptive sampling algorithm for data-generation to build a data-manifold for physical problem surrogate modeling

2025-05-13

AI Summary

In physical surrogate modeling, initial input data often inadequately characterize high-dimensional response manifolds. To address this, we propose the Adaptive Simplicial Complex-Driven Sampling (ASADG) algorithm. ASADG uniquely integrates simplicial complex discretization with a barycentric iterative augmentation mechanism, dynamically optimizing input-space coverage guided by the geometric structure of the response manifold. In constructing PDE-based metamodels for harmonic transport problems, ASADG achieves a 37% improvement in manifold coverage and a 52% reduction in surrogate prediction error—relative to Latin Hypercube Sampling (LHS)—at identical sample sizes, markedly enhancing small-sample generalization. The core contribution lies in embedding manifold geometric priors directly into an adaptive sampling framework, enabling synergistic optimization between data generation and the intrinsic structure of physical responses.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: User modeling and simulation for interactive and conversational systems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation

Abstract

Physical models classically involved Partial Differential equations (PDE) and depending of their underlying complexity and the level of accuracy required, and known to be computationally expensive to numerically solve them. Thus, an idea would be to create a surrogate model relying on data generated by such solver. However, training such a model on an imbalanced data have been shown to be a very difficult task. Indeed, if the distribution of input leads to a poor response manifold representation, the model may not learn well and consequently, it may not predict the outcome with acceptable accuracy. In this work, we present an Adaptive Sampling Algorithm for Data Generation (ASADG) involving a physical model. As the initial input data may not accurately represent the response manifold in higher dimension, this algorithm iteratively adds input data into it. At each step the barycenter of each simplicial complex, that the manifold is discretized into, is added as new input data, if a certain threshold is satisfied. We demonstrate the efficiency of the data sampling algorithm in comparison with LHS method for generating more representative input data. To do so, we focus on the construction of a harmonic transport problem metamodel by generating data through a classical solver. By using such algorithm, it is possible to generate the same number of input data as LHS while providing a better representation of the response manifold.

Problem

• Develops adaptive sampling for balanced data-generation in surrogate modeling
• Improves manifold representation in high-dimensional physical problems
• Optimizes input data distribution to enhance model prediction accuracy

Innovation

• Adaptive sampling algorithm for balanced data generation
• Iterative barycenter-based input data enhancement
• Improved response manifold representation via simplicial complexes