Approximating Young Measures With Deep Neural Networks

TL;DR

This work addresses non-convex variational problems by approximating gradient Young measures with a neural-network–based push-forward of a Gaussian. By representing each $ν_x$ as $ν_x=(f_x)_#γ$ and parameterizing $f_x$ with a ResNet, the authors turn the measure-valued relaxation into a tractable, data-driven optimization problem $(L(θ))$ that preserves the necessary curl-free gradient structure. They demonstrate the method on four non-convex cases, recovering key features of the expected Young measures (e.g., atoms at $±1$ and zero within the interior), and show convergence with respect to energy gaps through Wasserstein-distance–type controls. The approach offers a practical pathway to model microstructure formation in materials and related non-cooperative optimization problems, with code available for reproducibility and potential extensions to vector-valued fields and more complex multi-well energies.

Abstract

Parametrized measures (or Young measures) enable to reformulate non-convex variational problems as convex problems at the cost of enlarging the search space from space of functions to space of measures. To benefit from such machinery, we need powerful tools for approximating measures. We develop a deep neural network approximation of Young measures in this paper. The key idea is to write the Young measure as push-forward of Gaussian measures, and reformulate the problem of finding Young measures to finding the corresponding push-forward. We approximate the push-forward map using deep neural networks by encoding the reformulated variational problem in the loss function. After developing the framework, we demonstrate the approach in several numerical examples. We hope this framework and our illustrative computational experiments provide a pathway for approximating Young measures in their wide range of applications from modeling complex microstructure in materials to non-cooperative games.

Figures (8)

• Figure 1: Schematic of the neural network architecture and computation pipeline.
• Figure 2: (a) Convergence of total neural network loss during training. (b) Scalar field U obtained via cumulative integration of weighted gradients. (c) 3D surface of $F$ over $(x, \xi)$. (d) The push-forward map $\partial F/\partial \xi$ over $(x, \xi)$. (e) Distribution of the approximated gradient Young measure.
• Figure 3: Top row: (a) Convergence of the neural network’s total loss in log scale. (b) Predicted scalar field $u(x,y)$. Bottom row shows the neural network predictions of the field $F(\xi,\tau)$ at three representative points in the unit square $(x,y)=(0.5,0.5),(0.25,0.75),(0.75,0.25)$, respectively for (c)–(e).
• Figure 4: Top two rows: Components of the two dimensional $\nabla F$ as the push-forward map, depicted respectively at three representative points $(x,y)=(0.5,0.5),(0.25,0.75),(0.75,0.25)$. Bottom two rows: gradient Young measure densities obtained as push-forward of a Gaussian obtained as histograms using 10,000 Gaussian samples.
• Figure 5: Top row: (a) Convergence of the neural network’s total loss in log scale. (b) Predicted scalar field $u(x,y)$. Bottom row shows the neural network predictions of the field $F(\xi,\tau)$ at three representative points in the unit square $(x,y)=(0.5,0.5),(0.25,0.75),(0.75,0.25)$, respectively for (c)–(e).

Theorems & Definitions (3)

• Theorem 1
• Theorem 2
• Example 3.1