Courant: a State-Adaptive Perceiver-Based Neural Surrogate with Local Support and Interpretable Field Decomposition

2026-05-24

AI Summary

Traditional scientific machine learning models struggle to effectively capture the multiscale structure and dynamic evolution of complex physical fields due to their lack of local support, adaptivity, and interpretability. This work proposes Courant—an end-to-end neural surrogate model based on the Perceiver architecture—that introduces, for the first time, the concept of adaptive hp-refinement into neural modeling. By leveraging state-adaptive latent queries, shared random Fourier feature embeddings, and a lightweight decoder, Courant achieves spatially localized support and adaptive refinement in physical space. The latent variables are geometrically anchored, multiscale-specialized, and temporally consistent, enabling interpretable field decomposition. Experiments demonstrate that Courant attains competitive accuracy in both steady-state and transient simulations while automatically capturing multiscale geometric features and coherent dynamic structures.

Abstract

We introduce "Courant", a Perceiver-based encoder-processor-decoder surrogate model that has latent features exhibiting adaptive specialization and local support in the physical space, enabling functionality akin to an adaptive hp-refinement scheme, an attribute that is highly desirable in traditional numerical solvers and scientific machine learning broadly. The proposed architecture combines a shared random Fourier feature coordinate embedding, state-adapted latent queries, and a light-weight decoder. Courant is trained end-to-end with steady or transient simulation data and only a standard L_2 prediction loss in the physical space, achieving competitive accuracy on benchmarks. We demonstrate that Courant's inductive biases yield latents that are interpretable by design: they develop multiscale geometric specialization in the simulation domain and track coherent structures in the time-dependent case, acting analogously to time-evolving spatial basis functions and allowing for decoding a compact, geometry-anchored, partition-of-unity-like decomposition of the simulated field.

Problem

• neural surrogate
• local support
• interpretable decomposition
• adaptive specialization
• scientific machine learning

Innovation

• state-adaptive latent queries
• local support
• interpretable field decomposition
• Perceiver-based surrogate
• multiscale geometric specialization