Scientific machine learning: bridging the gap between theory and practice in deep learning for computational science and engineering applications

科学计算对现代生活的各个领域都是必不可少的。无论是预测极端天气事件，检测和模拟病毒爆发，还是生成高保真医学图像以更好地促进诊断，科学计算算法都可以为影响人们的关键决策和政策提供信息，并帮助创造更公平，更公正的社会。在接下来的五年里，通过结合机器学习（ML）技术，将有一个前所未有的机会快速推进这一领域。ML有望对科学计算及其许多应用产生重大影响。然而，这也带来了挑战。从理论的角度来看，ML技术的理解很少，并且越来越多的人担心现有技术在鲁棒性和可靠性方面还不符合该领域传统的严格标准。这项研究计划的总体目标是应对这些挑战。其长期愿景是帮助实现ML作为科学计算及其许多应用的范式改变工具的成功整合。具体而言，其目标是：1）开发强大，高保真，计算效率高，理论上有保证的端到端程序，用于基于神经网络和深度学习的复杂环境成像。这项工作预计将带来对目前最先进的方法的重大改进，在关键成像模式，如光场成像（例如电光和红外系统，高光谱成像，无透镜成像），医学成像（例如MRI，X射线CT）和科学成像（例如电子显微镜，无线电干涉测量）中产生切实的好处。2)为大规模、高维近似设计新的数据驱动技术，利用科学数据集中的低维结构来显着提高性能。这项工作将带来更好的方法，并在重要的科学计算任务中带来重大利益，例如不确定性量化和复杂系统的数据驱动发现。3)通过提供稳定性和准确性的理论上界和下界，为ML在科学计算中建立新的理论基础。这项工作将通过更好地理解科学计算问题中学习的理论极限来推进知识，为研究界未来的算法发展提供关键基准。该计划将在国家重要的关键领域为许多HQP提供培训。它将开发新的、创新的ML算法，适用于计算科学和工程中的一系列不同问题，并通过新的理论基础来推进科学计算中强大和可靠的ML知识。它将加速这一新兴但发展迅速的领域的进展，发展加拿大的关键领导地位，并刺激加拿大各行业的重要技术改进。

Scientific computing is essential to all areas of modern life. Whether it be forecasting extreme weather events, detecting and modelling virus outbreaks or producing high-fidelity medical images to better facilitate diagnoses, algorithms for scientific computing inform key decisions and policies that impact people and help create fairer, more equitable societies. Over the next five years, there is an unprecedented opportunity to rapidly advance this field through the incorporation of Machine Learning (ML) techniques. ML is poised to have a substantial impact on scientific computing and its many applications. Yet it also raises challenges. ML techniques are poorly understood from a theoretical perspective, and there is growing concern that existing techniques do not yet meet the traditional, rigorous standards of the field in terms of robustness and reliability. The overarching goal of this research program is to tackle these challenges. Its long-term vision is to help achieve the successful incorporation of ML as a paradigm-altering tool for scientific computing and its many applications. Specifically, its objectives are: 1) To develop robust, high-fidelity, computationally efficient and theoretically guaranteed end-to-end procedures for imaging complex environments based on neural networks and deep learning. This work is expected to bring significant improvements over current state-of-the-art methods, yielding tangible benefits in key imaging modalities such as light-field imaging (e.g. electro-optical and infrared systems, hyperspectral imaging, lensless imaging), medical imaging (e.g. MRI, X-Ray CT) and scientific imaging (e.g. electron microscopy, radio interferometry). 2) To design new data-driven techniques for large-scale, high-dimensional approximation that leverage low-dimensional structure in scientific datasets to significantly enhance performance. This work will lead to better methods and bring significant benefits in important scientific computing tasks such as uncertainty quantification and data-driven discovery of complex systems. 3) To establish new theoretical foundations for ML in scientific computing by providing both theoretical upper and lower bounds for stability and accuracy. This work will advance knowledge through a better understanding of the theoretical limits of learning in scientific computing problems, providing key benchmarks for future algorithmic developments by the research community. This program will provide training for many HQP in key areas of national importance. It will develop new, innovative ML algorithms that are applicable across a range of different problems in computational science and engineering, and advance knowledge through new theoretical foundations for robust and reliable ML in scientific computing. It will accelerate progress in this nascent, but rapidly developing field, develop key Canadian leadership and stimulate important technological improvements across a range of Canadian industries.