提出可微分的多阱势能模型，自动发现模式数与跃迁尺度。
Differentiable neural network representation of multi-well, locally-convex potentials
- 用LSE-ICNN构建平滑且凸的多阱势能表示
- 通过稀疏回归自动确定模式数量与跃迁尺度
- 适用于相变、基因调控等复杂多模系统建模
多阱势能在物理、化学和生物学中广泛存在，用于描述相变、动态不稳定性及多模态行为。本文提出一种基于输入凸神经网络（ICNN）的对数求和指数（LSE）混合模型，实现可微分且局部凸的势能表示。该方法在势阱内保持凸性，支持梯度学习与推断。关键优势在于通过稀疏回归自动识别模式数量与跃迁尺度，实现自适应、简洁建模。我们在机械化学相变、微观结构弹性失稳、保守型生物基因回路及多模态概率分布的变分推断等多个领域验证了该模型的通用性，证明其在保留可微性的同时有效捕捉复杂多模态景观，适用于数据驱动建模、优化与物理仿真。
原文摘要 · Abstract (English)
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.
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