Man, Machine, and Mathematics: A mathe-physical search for differential equation solutions via deep learning

人、机器和数学：通过深度学习对微分方程解进行数学物理搜索

• 批准号: 2602638
• 金额: --
• 依托单位: Imperial College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2602638
• 关键词: Man; Machine; Mathematics; mathe; physical

Well posed mathematical problems are seldom solvable in an explicit, conventional sense. A notable example of this phenomenon is within the field of differential equations, where an exact solution often exists but is analytically impossible to find. The development of approximation methods to estimate such well-parametrized but not explicitly knowable solutions is an important endeavour within the field.This project presents a blueprint for building such approximations via deep learning. We build our methods on assumptions satisfied by a large class of differential equations, such as Frechet differentiability of the equation operators and compact domains of interest. We aim to demonstrate that neural network solution models are almost always capable of being found under such assumptions. We also hope to present explicit results on the errors expected from these models, alongside techniques for quantifying and minimising those errors. Finally, we aim to provide strict guarantees on the model sizes, architectures, optimization run-times, etc., needed to search for models that are accurate up to pre-specified tolerances.The project will hope to advance the field of neural network-based differential equation solving by adding two novel facets to it. First, by fusing ideas from numerical methods and error analysis into deep learning of parametrized objects, we hope to move the field away from its reliance on surrogate markers, like loss functions, as a means of error analysis. In turn, that will also lead us to methods of efficient error correction.Second, we aim to provide a rigorous set of a priori-decidable strategies for efficient model building by leveraging the emerging advances in computing infrastructure and the algorithms that can tap into them.Indeed, the project has already seen its first successes in the modelling of dynamical systems' differential equations (Physical Review E 105, 065305) and in sparsifying deep networks (Redman et al., ICML 2022). Given the generality of the results developed in the latter work (it concerns all deep networks, not just differential equation solution models), we believe the project will lead to significant results beyond its originally planned scope.Success in these objectives will require the creation of new techniques within the fields of random dynamical systems, stochastic computational methods, deep learning methods for equation solving, and optimization/complexity analysis, amongst others, each of which plays a central role in the CDT's research and training missions.

好的数学问题很少能用明确的、传统的意义来解决。这种现象的一个显著的例子是在微分方程领域中，在这个领域中，精确解经常存在，但却无法通过解析找到。发展近似方法来估计这类参数化良好但不显式可知的解是该领域的一项重要努力。这个项目展示了一个通过深度学习构建这种近似的蓝图。我们的方法建立在一大类微分方程所满足的假设上，例如方程算子的Frechet可微性和感兴趣的紧定义域。我们的目标是证明神经网络解模型几乎总是能够在这样的假设下被发现。我们还希望对这些模型的预期误差给出明确的结果，以及量化和最小化这些误差的技术。最后，我们的目标是在模型尺寸、架构、优化运行时间等方面提供严格的保证，以搜索精确到预先指定公差的模型。该项目希望通过增加两个新的方面来推进基于神经网络的微分方程求解领域。首先，通过将数值方法和误差分析的思想融合到参数化对象的深度学习中，我们希望将该领域从依赖替代标记（如损失函数）作为误差分析的手段中移开。反过来，这也将引导我们找到有效的纠错方法。其次，我们的目标是通过利用计算基础设施和可以利用它们的算法的新兴进展，为有效的模型构建提供一套严格的优先级可决定策略。事实上，该项目已经在动力系统微分方程建模（Physical Review E 105, 065305）和深度网络稀疏化（Redman et al., ICML 2022）方面取得了第一次成功。鉴于后一项工作中开发的结果的通用性（它涉及所有深度网络，而不仅仅是微分方程解模型），我们相信该项目将导致超出其最初计划范围的重要结果。这些目标的成功将需要在随机动力系统、随机计算方法、求解方程的深度学习方法和优化/复杂性分析等领域创造新技术，其中每一个都在CDT的研究和培训任务中发挥着核心作用。