This paper presents Mechanistic Neural Networks, a neural network design for machine learning applications in the sciences. It incorporates a new Mechanistic Block in standard architectures to explicitly learn governing differential equations as representations, revealing the underlying dynamics of data and enhancing interpretability and efficiency in data modeling. Central to our approach is a novel Relaxed Linear Programming Solver (NeuRLP) inspired by a technique that reduces solving linear ODEs to solving linear programs. This integrates well with neural networks and surpasses the limitations of traditional ODE solvers enabling scalable GPU parallel processing. Overall, Mechanistic Neural Networks demonstrate their versatility for scientific machine learning applications, adeptly managing tasks from equation discovery to dynamic systems modeling. We prove their comprehensive capabilities in analyzing and interpreting complex scientific data across various applications, showing significant performance against specialized state-of-the-art methods.

翻译：本文提出了一种面向科学机器学习应用的神经网络设计——机理神经网络。该设计在标准架构中引入了全新的机理模块，以显式学习作为表征的控制微分方程，揭示数据的底层动态过程，提升数据建模的可解释性与效率。我们的核心方法是一种受技术启发的新型松弛线性规划求解器（NeuRLP），该技术将求解线性常微分方程转化为求解线性规划问题。该求解器与神经网络无缝集成，突破了传统常微分方程求解器的局限，支持可扩展的GPU并行处理。总体而言，机理神经网络展示了其在科学机器学习应用中的多面性，能够熟练处理从方程发现到动态系统建模等任务。我们通过多种应用验证了其在分析和解释复杂科学数据方面的全面能力，在与专门的先进方法对比中展现出显著性能优势。