Mallat Scattering Transformation based surrogate for MagnetoHydroDynamics (2302.10243v2)

Abstract: A Machine and Deep Learning methodology is developed and applied to give a high fidelity, fast surrogate for 2D resistive MHD simulations of MagLIF implosions. The resistive MHD code GORGON is used to generate an ensemble of implosions with different liner aspect ratios, initial gas preheat temperatures (that is, different adiabats), and different liner perturbations. The liner density and magnetic field as functions of $x$, $y$, and $t$ were generated. The Mallat Scattering Transformation (MST) is taken of the logarithm of both fields and a Principal Components Analysis is done on the logarithm of the MST of both fields. The fields are projected onto the PCA vectors and a small number of these PCA vector components are kept. Singular Value Decompositions of the cross correlation of the input parameters to the output logarithm of the MST of the fields, and of the cross correlation of the SVD vector components to the PCA vector components are done. This allows the identification of the PCA vectors vis-a-vis the input parameters. Finally, a Multi Layer Perceptron neural network with ReLU activation and a simple three layer encoder/decoder architecture is trained on this dataset to predict the PCA vector components of the fields as a function of time. Details of the implosion, stagnation, and the disassembly are well captured. Examination of the PCA vectors and a permutation importance analysis of the MLP show definitive evidence of an inverse turbulent cascade into a dipole emergent behavior. The orientation of the dipole is set by the initial liner perturbation. The analysis is repeated with a version of the MST which includes phase, called Wavelet Phase Harmonics (WPH). While WPH do not give the physical insight of the MST, they can and are inverted to give field configurations as a function of time, including field-to-field correlations.

• The paper introduces an MST-based surrogate that significantly reduces computational costs for simulating 2D resistive MHD dynamics.
• The approach uses PCA and SVD to extract key features from simulation data, capturing 94% of the variance for high-fidelity reconstructions.
• The study reveals an inverse turbulent cascade and emergent dipole behavior, offering valuable insights into complex plasma interactions.

Mallat Scattering Transformation-Based Surrogate for MagnetoHydroDynamics

The paper investigates a novel Machine and Deep Learning (MLDL) approach employing the Mallat Scattering Transformation (MST) to create a surrogate model for 2D resistive MagnetoHydroDynamics (MHD) simulations, particularly for Magnetic Liner Inertial Fusion (MagLIF) implosions. Utilizing the resistive MHD code GORGON, the paper generates an ensemble of simulations with varying liner aspect ratios, initial gas preheat temperatures, and liner perturbations. The research aims to reduce computational costs associated with high-fidelity MHD simulations by developing a rapid, high-fidelity surrogate model encapsulating the essential emergent behaviors and correlations of the simulated processes.

Methodology Overview

The methodology employs the MST, an effective dimensionality reduction tool akin to a convolutional neural network with predefined weights, which is robust to diffeomorphic deformations and possesses translational invariance. The MST is applied to the logarithm of the liner densities and magnetic fields from the simulation outputs. A Principal Components Analysis (PCA) is subsequently performed to reduce the dimensionality of the MST representation, extracting the most significant components that capture the primary variation in simulation data. Singular Value Decomposition (SVD) is employed to correlate these PCA components back to the input parameters, facilitating a physical interpretation of the data attributes.

Machine Learning Implementation

A Multi Layer Perceptron (MLP) neural network with ReLU activation and a straightforward three-layer encoder/decoder architecture is trained to predict the PCA components of the fields over time. Remarkably, the PCA components account for 94% of the variance, underscoring the efficiency of this technique. The network predicts these components with high accuracy, enabling the reconstruction of high-fidelity field dynamics from limited computational resources.

Dynamic Behavior and Insights

The paper highlights evidence of an inverse turbulent cascade in the liner dynamics, culminating in a dipole emergent behavior. The phase of this dipole mode is determined by the initial perturbations, indicating a self-organization within the turbulent flow—an insight that could have profound implications for understanding the nonlinear dynamics in plasma systems. The use of MST and an extended version incorporating phase (Wavelet Phase Harmonics, WPH) further helps to reveal field-to-field correlations, unveiling complex interactions within these dynamic systems.

Implications and Future Directions

Practically, this research offers a pathway to conduct rapid simulations for MHD applications, drastically lowering the computational expense while maintaining high fidelity. Theoretically, it suggests that reduced-order modeling techniques anchored on physics-informed transformations like the MST could provide deeper analytical insights into emergent and self-organized phenomena within complex systems. Future developments could explore extending the methodology to 3D simulations, analyzing the implications of helicity conservation in 3D plasma MHD, or refining the transformation techniques to optimize their performance and interpretability. Moreover, testing the extrapolative capabilities of these models could lead to enhanced predictive abilities in new MHD regimes, contributing to advancements in experimental design and theoretical modeling within the field.

This paper illustrates a critical stride in using machine learning to emulate complex physical systems, blending computational efficiency with the pursuit of fundamental insights into nonlinear dynamics and emergent behavior in MHD contexts.