Scientific Machine Learning
Describe the theoretical properties of methods based on machine learning to solve forward, inverse and operator design problems associated with partial differential equations.
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Recent advances in the application of machine learning techniques to solve scientific problems have turned them into important tools in numerical analysis, specially in the context of numerically solving forward, inverse and operator design problems associated with partial differential equations (PDEs). Methods such as the physics-informed neural networks and the deep operator networks have been recently proposed to tackle numerical analysis problems. Even though they are being widely applied to solve important practical problems, there is a lack of mathematical understanding about them that makes this a fruitful research area for mathematicians.
Projects will involve studying theoretical properties about the approximation of functions and operators by neural networks, learning algorithms, and statistical properties of machine learning methods in this context. The theoretical results are expected to lead to improved practical methods that will be empirically compared with state-of-the-art methods in scientific machine learning with applications in engineering and physics modelling.
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MSI-Google Fellow