Generative Learning of Densities on Manifolds

Research Summary

This study introduces and validates a framework for generating data distributions on high-dimensional manifolds using modified score-based generative models (m-SGMs). The model leverages nonlinear dimensionality reduction methods to address challenges in high-dimensional uncertainty quantification. Applications in computational mechanics demonstrate the robustness and efficiency of the proposed approach over conventional methods.

Motivation

High-dimensional data encountered in multiscale computational mechanics, like stress-strain relationships in complex materials, presents challenges for accurate and efficient modeling. The framework developed in this paper aims to address:

1. The curse of dimensionality in simulations and sampling strategies.
2. Generative modeling of data constrained to manifolds defined by experimental and synthetic results.
3. Incorporation of physics-based features in generative approaches to ensure consistency with observed phenomena.

By employing manifold learning and score-based techniques, the paper seeks to improve the fidelity of synthetic data generation and its applicability to real-world computational mechanics problems.

Methodology

Framework Details

1. Modified Score-Based Generative Models (m-SGMs):

   • Adapt score-based algorithms to directly model multiscale material systems.
   • Integrate physics-based parameters to maintain correspondence with real data.

2. Double Diffusion Maps:

   • Utilize these as a dimensionality reduction tool to parameterize manifolds, facilitating sampling in high-dimensional spaces.

3. KL Divergence Performance Metrics:

   • Statistical comparison of the generated distributions against ground truth to quantify model precision.

Dataset Used

• 800-point simulation comparing tensile and bending test outcomes, structured into 449 feature dimensions.
• Dimensions include microscale, mesoscale, and macroscale properties.

Techniques Applied

To generate additional realizations for high-dimensional synthetic datasets, the framework employs m-SGMs combined with double diffusion maps, surpassing typical generation techniques in efficiency and quality.

Results

Applications

1. Synthetic Stress-Strain Data:

   • Generated distributions accurately mimic observed stress-strain curves from laboratory tests.
   • Validated results include accurate reproductions under bending and tensile configurations.

2. Dimensionality Reduction:

   • Experimental data constrained within manifolds showed strong agreement with real-world outcomes.
   • Double diffusion maps provided significant enhancements to sampling precision.

3. Performance Analysis:

   • Among methods evaluated, m-SGM1 produced the lowest KL divergence, indicating the highest fidelity to true data distributions.

Computational Metrics

• Improved runtime performance relative to standard Monte Carlo Simulations (MCS).
• Increased sampling precision across multiscale dimensional embeddings.

Key Insights

• Generative Modeling vs. MCS:
  • Modified approaches reduce computational complexity while improving fidelity.
• Physics-Based Integration:
  • Ensures synthetic data aligns with physical constraints of observed phenomena.
• Scalability:
  • Applicability expands to larger, complex datasets by leveraging manifold-parametrization techniques.

Future Work

1. Expanding to stochastic dynamics and chaotic systems.
2. Improving framework scalability for higher-dimensional contexts.
3. Creating publicly accessible benchmarks for generative models designed for physical systems modeling.