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Physics- and data-driven Active Learning of neural network representations for free energy functions of materials from statistical mechanics

Jamie Holber, Krishna Garikipati

TL;DR

The paper tackles the challenge of building accurate, atomistically informed free energy representations for materials by learning a differentiable surrogate of the free energy density $g(\vec{\eta})$ from derivative data $\vec{\mu}=\partial g/\partial \vec{\eta}$ obtained via DFT-informed Monte Carlo. It integrates an Integrable Deep Neural Network (IDNN) with symmetry-invariant features to ensure physically consistent energy landscapes, and couples this with a versatile Active Learning workflow that blends global space-filling sampling, physics-driven targeted sampling (wells, endpoints, regions of instability, and lowest-energy paths), and uncertainty-based criteria (High Error, Highly Sensitive Regions, QBC) plus novelty and dynamic hyperparameter reweighting. The framework demonstrates how different sampling strategies impact global and regional accuracy, showing that combinations like 7D plus limited 2D sampling and QBC or novelty-enhanced approaches can substantially reduce the required data while preserving fidelity in physically relevant regions. The work offers a robust, scalable methodology for Monte Carlo sampling of high-dimensional free energy landscapes, with direct applicability to phase-field modeling and materials design. The practical impact lies in enabling efficient, principled generation of atomistically informed free energy models for complex materials systems, reducing computational cost and improving predictive reliability in phase dynamics simulations.

Abstract

Accurate free energy representations are crucial for understanding phase dynamics in materials. We employ a scale-bridging approach to incorporate atomistic information into our free energy model by training a neural network on DFT-informed Monte Carlo data. To optimize sampling in the high-dimensional Monte Carlo space, we present an Active Learning framework that integrates space-filling sampling, uncertainty-based sampling, and physics-informed sampling. Additionally, our approach includes methods such as hyperparameter tuning, dynamic sampling, and novelty enforcement. These strategies can be combined to reduce MSE,either globally or in targeted regions of interest,while minimizing the number of required data points. The framework introduced here is broadly applicable to Monte Carlo sampling of a range of materials systems.

Physics- and data-driven Active Learning of neural network representations for free energy functions of materials from statistical mechanics

TL;DR

The paper tackles the challenge of building accurate, atomistically informed free energy representations for materials by learning a differentiable surrogate of the free energy density from derivative data obtained via DFT-informed Monte Carlo. It integrates an Integrable Deep Neural Network (IDNN) with symmetry-invariant features to ensure physically consistent energy landscapes, and couples this with a versatile Active Learning workflow that blends global space-filling sampling, physics-driven targeted sampling (wells, endpoints, regions of instability, and lowest-energy paths), and uncertainty-based criteria (High Error, Highly Sensitive Regions, QBC) plus novelty and dynamic hyperparameter reweighting. The framework demonstrates how different sampling strategies impact global and regional accuracy, showing that combinations like 7D plus limited 2D sampling and QBC or novelty-enhanced approaches can substantially reduce the required data while preserving fidelity in physically relevant regions. The work offers a robust, scalable methodology for Monte Carlo sampling of high-dimensional free energy landscapes, with direct applicability to phase-field modeling and materials design. The practical impact lies in enabling efficient, principled generation of atomistically informed free energy models for complex materials systems, reducing computational cost and improving predictive reliability in phase dynamics simulations.

Abstract

Accurate free energy representations are crucial for understanding phase dynamics in materials. We employ a scale-bridging approach to incorporate atomistic information into our free energy model by training a neural network on DFT-informed Monte Carlo data. To optimize sampling in the high-dimensional Monte Carlo space, we present an Active Learning framework that integrates space-filling sampling, uncertainty-based sampling, and physics-informed sampling. Additionally, our approach includes methods such as hyperparameter tuning, dynamic sampling, and novelty enforcement. These strategies can be combined to reduce MSE,either globally or in targeted regions of interest,while minimizing the number of required data points. The framework introduced here is broadly applicable to Monte Carlo sampling of a range of materials systems.

Paper Structure

This paper contains 25 sections, 16 equations, 11 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Background/Motivation
  3. Symmetrically Equivalent Order Parameters
  4. Monte Carlo
  5. Symmetry-invariant linear combinations of order parameters
  6. Integrable Deep Neural Network (IDNN)
  7. Active Learning
  8. Overview of Active Learning Workflow
  9. Criteria for Sampling
  10. Benchmark: global sampling
  11. Known Areas of Interest: Wells/End members/vertices
  12. High Error
  13. Highly Sensitive Regions
  14. Query by Committee
  15. Locating Free Energy Wells
  16. ...and 10 more sections

Figures (11)

  • Figure 1: (a) The phase diagram was constructed using a tangent line approach based on the 1D IDNN at various temperatures with zigzag ordering. (b) The 12 variants corresponding to the zigzag ordering. The highlighted boxes on each variant show the smallest supercell which captures the details of the ordering. These three supercells were used to construct a 32-site mutually commensurate supercell teichert2023bridging
  • Figure 2: The results of Monte Carlo simulations varying Lithium composition at 300 K with and without bias sampling (umbrella sampling).
  • Figure 3: These figures show sample surfaces/data to illustrate regions of importance for our Active Learning workflow. (a) Illustration of a sample free energy surfaces. The energy wells are circled in red, and a region of non-convexity is circled in black. (b) A dataset (blue points) and its fit (green line). High-error points, where the fit is worst, are circled in red. A region with a large gradient (high sensitivity) is circled in black. (c) A dataset (blue points) with different fits (three curves). Query by Committee samples additional points at the red dotted lines where the fits differ most.
  • Figure 4: The results of the Active Learning workflow for Billiard walk 7D sampling only (Billiard walk 7D), Billiard walk sampling 7D and 2D (Billiard walk 7D + 2D) and Billiard walk sampling 7D and 2D with High Error points (High Error). Mean Squared Error (MSE) vs. round for (a) the final training set (from round 10) and (b) the testing set. High Error has a lower MSE for testing compared to training due to the choice for data in the testing set as discussed in the text.
  • Figure 5: The results of the Active Learning workflow based on different criteria are shown. All results use Billiard walk sampling in 7D and 2D with high-error points. The High Error line is the same as in Figure \ref{['fig:mse_comparison1']}, while the other lines incorporate the additional criteria specified by their names. Mean Squared Error (MSE) vs. round for (a) the final training set (from round 10) and (b) the testing set. QBC has a lower MSE for testing compared to training due to the choice for data in the testing set as discussed in the text.
  • ...and 6 more figures