Learning discrete Lagrangians for variational PDEs from data and detection of travelling waves
Christian Offen, Sina Ober-Blöbaum
发表信息：
Learning discrete Lagrangians for variational pdes from data and detection of travelling waves. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. vol. 14071, pp. 569-579, Springer Nature Switzerland, Cham (2023)
The article shows how to learn models of dynamical systems from data which are governed by an unknown variational PDE. Rather than employing reduction techniques, we learn a discrete field theory governed by a discrete Lagrangian density $L_d$ that is modelled as a neural network. Careful regularisation of the loss function for training $L_d$ is necessary to obtain a field theory that is suitable for numerical computations: we derive a regularisation term which optimises the solvability of the discrete Euler--Lagrange equations. Secondly, we develop a method to find solutions to machine learned discrete field theories which constitute travelling waves of the underlying continuous PDE.
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Numerical simulation is indispensable in industrial design processes. It can replace expensive experiments and even reduce the need for prototypes. While products designed with the aid of numerical simulation undergo continuous improvement, this must also be true for numerical simulation itself. Up to date, no general purpose numerical method is available which can accurately resolve a variety of physics ranging from fluid to solid mechanics including large deformations and free surface flow phenomena. These complex multi-physics problems occur for example in Additive Manufacturing processes. …
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