Hierarchical-embedding autoencoder with a predictor (HEAP) as efficient architecture for learning long-term evolution of complex multi-scale physical systems

Abstract: We propose a novel efficient architecture for learning long-term evolution in complex multi-scale physical systems which is based on the idea of separation of scales. Structures of various scales that dynamically emerge in the system interact with each other only locally. Structures of similar scale can interact directly when they are in contact and indirectly when they are parts of larger structures that interact directly. This enables modeling a multi-scale system in an efficient way, where interactions between small-scale features that are apart from each other do not need to be modeled. The hierarchical fully-convolutional autoencoder transforms the state of a physical system not just into a single embedding layer, as it is done conventionally, but into a series of embedding layers which encode structures of various scales preserving spatial information at a corresponding resolution level. Shallower layers embed smaller structures on a finer grid, while deeper layers embed larger structures on a coarser grid. The predictor advances all embedding layers in sync. Interactions between features of various scales are modeled using a combination of convolutional operators. We compare the performance of our model to variations of a conventional ResNet architecture in application to the Hasegawa-Wakatani turbulence. A multifold improvement in long-term prediction accuracy was observed for crucial statistical characteristics of this system.

Hierarchical-Embedding Autoencoder with a Predictor (HEAP) for Modeling Complex Multi-Scale Physical Systems

This paper introduces the Hierarchical-Embedding Autoencoder with a Predictor (HEAP), an advanced architectural model designed for the efficient simulation of complex multi-scale physical systems. Traditionally, modeling such systems presents significant challenges due to the need to account for interactions at different scales while maintaining computational feasibility. The authors of this study address this issue through a novel approach that leverages hierarchical embedding and convolutional architectures.

Core Contributions and Methodology

HEAP enhances the conventional Fully-Convolutional Autoencoder (FCAE) structures by incorporating a hierarchy of embeddings for different scales within the physical system. The primary advancement lies in the subdivision of embedding layers, each corresponding to a distinct scale. This multidimensional approach ensures spatial information preservation at various resolution levels and optimally represents the physical structures of changing scale.

The model operates by embedding smaller structures in shallower layers while accommodating larger structures in deeper layers. Interactions between different scale features are efficiently modeled using convolutional operators, facilitating an increased level of precision compared to single-layer embeddings used in traditional models.

The efficacy of HEAP is benchmarked against variations of conventional ResNet architectures, specifically through its application in modeling Hasegawa-Wakatani turbulence — a representative case study relevant for predicting long-term behavior in magnetized plasma turbulence. The study exhibits multifold improvements in prediction accuracy for significant statistical characteristics of this turbulent system, highlighting the model's effectiveness.

Numerical Results and Analysis

The HEAP architecture presented in the paper is supported by robust numerical experiments comparing its performance against conventional FCAE with ResNet-based predictors, along with deeper convolutional models (C1-C3). Notably, hierarchical models with at least two layers (H2-H5) were found to outperform single-layer (conventional) models, particularly in reproducing complex fields accurately and improving representations of high-frequency details in statistical predictions.

The spatial and temporal Fourier spectra analyses reflect HEAP's superior capability in representing the modeled system, especially in generating statistically coherent solutions over long-time model rollouts. This is particularly crucial for turbulent systems where precise modeling of emergent phenomena, such as vortices, necessitates capturing long-range interactions effectively — a task HEAP demonstrates through notable accuracy improvements.

Practical and Theoretical Implications

HEAP’s innovations contribute significantly to the efficiency and precision of modeling multi-scale systems, making it suitable for applications requiring computational performance gains alongside reduced complexity. The hierarchical approach not only enhances prediction accuracy but reduces training data requirements, affirming its relevance in environments constrained by computational and data resources.

The hierarchical prediction methodology, applied in embedding space, shows potential for scaling into higher-dimensional systems and adopting methods such as graph networks for unstructured data representations. Moreover, potential applications in fluid turbulence modeling and reduced-order kinetic models confirm the broader applicability of HEAP in both academic investigations and industrial applications where advanced simulation techniques are crucial.

Speculations on Future Directions

The HEAP architecture opens avenues for future developments, particularly in extending its principles to three-dimensional systems via 3D convolutions, thus broadening its impact in comprehensive physics-informed model development. Furthermore, potential enhancements integrating graph-based neural networks could introduce adaptive modeling capabilities, handling diverse datasets beyond structured grids.

Overall, the paper underscores the potential of hierarchical embeddings in advancing computational modeling of complex physical systems, marking a step towards more efficient and effective simulation architectures that cater to intricate scale hierarchies universe within multi-scale systems.