仓本神经网络算子：通过耦合振子动力学学习求解偏微分方程
The Kuramoto Neural Operator: Learning to Solve PDEs via Coupled Oscillator Dynamics
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中文总结 AI 辅助
本文提出仓本神经网络算子（KNO），通过交互振子潜场演化表示偏微分方程解，在PDE基准测试中性能优于竞争方法，其预测误差与潜振子同步程度相关。
中文摘要 AI 辅助
算子学习是计算科学中快速发展的领域，尤其适用于需在不同物理构型下反复求解偏微分方程（PDE）的问题。现有多数架构将解算子表示为固定基，该假设虽契合全局结构，但对物理空间中局部相互作用主导的现象适用性较差。本文受耦合振子系统的连续极限可描述广泛PDE类的观察启发，探索了一种替代视角，在此基础上提出仓本神经网络算子（KNO），通过交互振子的潜场演化表示解。在多样的PDE基准测试中，KNO实现了优异的预测性能，优于竞争方法；实验评估还包含广泛的 ablation 研究，量化了KNO各架构组件的贡献。此外，本文表明模型的预测误差与潜振子的集体动力学密切相关，其随振子同步程度呈系统性变化，为潜在机制提供了见解。
英文摘要
Operator learning is a rapidly advancing area of computational science. It is particularly well suited to problems where a partial differential equation (PDE) must be solved repeatedly under varying physical configurations. Most existing architectures represent the solution operator in a fixed basis. While this assumption is well aligned with global structures, it is less suitable for phenomena governed by local interactions in physical space. We explore an alternative perspective motivated by the observation that the continuum limit of coupled oscillator systems can describe a broad class of PDEs. Building on this idea, we introduce the Kuramoto Neural Operator (KNO), which represents the solution through the evolution of a latent field of interacting oscillators. Across a diverse collection of PDE benchmarks, KNO achieves strong predictive performance, with improvements over competing approaches. Our experimental evaluation also includes an extensive ablation study that quantifies the contribution of each architectural component incorporated into KNO. Furthermore, we show that the model's prediction error is closely linked to the collective dynamics of the latent oscillators. It varies systematically with their degree of synchronization, providing insights into the underlying mechanisms.
发表机构
- Basic Research of Artificial Intelligence Laboratory (BRAIn Lab)（人工智能基础研究实验室（BRAIn实验室））
- Innopolis University（Innopolis大学）
机构由 AI 辅助整理，请以论文原文为准。
