# Differentiable neural network representation of multi-well, locally-convex potentials

arXiv: https://arxiv.org/abs/2506.17242 · HTML: https://mlsanity.com/paper/2506.17242/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 68 references that are in the map.
- First posted on arXiv 6 Jun 2025.

## Builds on (most cited first, 18 of 18)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Optimal transport mapping via input convex neural networks](https://mlsanity.com/paper/1908.10962/) (2019)
- [On the Properties of the Softmax Function with Application in Game Theory and Reinforcement Learning](https://mlsanity.com/paper/1704.00805/) (2017)
- [Generalized Multimodal ELBO](https://mlsanity.com/paper/2105.02470/) (2021)
- [Optimal Control Via Neural Networks: A Convex Approach](https://mlsanity.com/paper/1805.11835/) (2018)
- [Polyconvex anisotropic hyperelasticity with neural networks](https://mlsanity.com/paper/2106.14623/) (2021)
- [Stacking for Non-mixing Bayesian Computations: The Curse and Blessing of Multimodal Posteriors](https://mlsanity.com/paper/2006.12335/) (2020)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling](https://mlsanity.com/paper/2109.11028/) (2021)
- [Neural network-based multiscale modeling of finite strain magneto-elasticity with relaxed convexity criteria](https://mlsanity.com/paper/2308.16311/) (2023)
- [CDiNN -Convex Difference Neural Networks](https://mlsanity.com/paper/2103.17231/) (2021)
- [Automated model discovery of finite strain elastoplasticity from uniaxial experiments](https://mlsanity.com/paper/2408.14615/) (2024)
- [Bridging scales with Machine Learning: From first principles statistical mechanics to continuum phase field computations to study order disorder transitions in LixCoO2](https://mlsanity.com/paper/2302.08991/) (2023)
- [Experiment-informed finite-strain inverse design of spinodal metamaterials](https://mlsanity.com/paper/2312.11648/) (2023)
- [Input Specific Neural Networks](https://mlsanity.com/paper/2503.00268/) (2025)
- [Uncertainty quantification of neural network models of evolving processes via Langevin sampling](https://mlsanity.com/paper/2504.14854/) (2025)

## Built on by (most cited first, 1 of 1)

- [LaGSplat: Inferring Physics-Governed Interactive Simulation from Monocular Video Using Latent Lagrangian Gaussian Splatting](https://mlsanity.com/paper/2608.16324/) (2026)

## Abstract (arXiv)

Multi-well potentials are ubiquitous in science, modeling phenomena such as phase transitions, dynamic instabilities, and multimodal behavior across physics, chemistry, and biology. In contrast to non-smooth minimum-of-mixture representations, we propose a differentiable and convex formulation based on a log-sum-exponential (LSE) mixture of input convex neural network (ICNN) modes. This log-sum-exponential input convex neural network (LSE-ICNN) provides a smooth surrogate that retains convexity within basins and allows for gradient-based learning and inference. A key feature of the LSE-ICNN is its ability to automatically discover both the number of modes and the scale of transitions through sparse regression, enabling adaptive and parsimonious modeling. We demonstrate the versatility of the LSE-ICNN across diverse domains, including mechanochemical phase transformations, microstructural elastic instabilities, conservative biological gene circuits, and variational inference for multimodal probability distributions. These examples highlight the effectiveness of the LSE-ICNN in capturing complex multimodal landscapes while preserving differentiability, making it broadly applicable in data-driven modeling, optimization, and physical simulation.