HAMNO: A Hierarchical Adaptive Multi-scale Neural Operator with Physics-Informed Learning for Dynamical Systems

2026-06-10

AI Summary

Existing neural operators struggle to effectively model nonlinear time-dependent dynamical systems characterized by multiscale structures, long-range interactions, and long-term stable evolution. This work proposes the Hierarchical Adaptive Multiscale Neural Operator (HAMNO), which integrates local convolutions with global spectral operators and employs a data-dependent gating mechanism to enable spatially adaptive fusion of local and global information. Furthermore, a physics-informed variant, PI-HAMNO, is introduced that uniquely combines strong-form PDE residuals with weak-form finite element test function constraints, enhanced by tetrahedral numerical integration to improve physical consistency and data efficiency. Experiments on the Allen–Cahn, Cahn–Hilliard, and Swift–Hohenberg equations demonstrate that HAMNO significantly outperforms existing baselines, while PI-HAMNO further enhances long-term prediction accuracy, stability, and generalization capability.

Abstract

Neural operators provide a powerful framework for learning solution mappings of partial differential equations directly in function space. However, many existing architectures still struggle to represent nonlinear time-dependent systems that involve multi-scale structures, long-range interactions, and stable long-time evolution. In this work, we introduce the Hierarchical Adaptive Multi-scale Neural Operator (HAMNO), a neural-operator architecture that combines local convolutional representations, global spectral operators, and hierarchical encoder-decoder processing. The central component of HAMNO is a data-dependent gating mechanism that adaptively balances local and global information at each spatial location, allowing the model to resolve fine-scale features while preserving long-range dependencies. We further develop a physics-informed extension, PI-HAMNO, based on a multi-objective loss strategy that combines data fitting with strong- and weak-form physics constraints. The strong-form term penalizes the domain-integrated squared PDE residual in physical coordinates, while the weak-form term is constructed by multiplying the governing residual by finite-element test functions and evaluating the resulting element integrals using centroid-based tetrahedral quadrature. The framework is evaluated on non-periodic Allen-Cahn (AC), Cahn-Hilliard (CH), and Swift-Hohenberg (SH) equations defined on cubic domains. Across long-horizon rollout, data-limited training, out-of-distribution initial-condition shifts, and random-seed variations, HAMNO improves predictive accuracy over standard neural-operator baselines, while PI-HAMNO further enhances stability, physical consistency, and data efficiency. The implementation is publicly available at https://github.com/MBamdad/HAMNO .

Problem

• multi-scale
• long-range interactions
• long-time evolution
• nonlinear dynamical systems
• neural operators

Innovation

• neural operator
• multi-scale modeling
• physics-informed learning
• adaptive gating
• hierarchical architecture