Physical laws and deep learning

Physical laws in our four dimensional world (time + 3D space) are usually mathematically abstracted in the form of partial differential equations. These equations are therefore broadly applied in various boundary value problems(BVPs) arising from natural phenomena or engineering practice, typically, such as wave propagation, heat transfer and elasticity problems. To solve these BVPs, analytical methods (e.g. seperation of variable method, eigenfunction expansion method, Laplace transform method) are limited to simple configurations. As the development of computer techniques, numerical methods (e.g. finite element method, finite volume method and finite difference method) show increasingly more powerful capability in solving these BVPs. A natural trend is the fution of the computer techniques and engineering. Thus, numerous engineering software companies (ANSYS, Dassault, Altair, MSC, etc.) emerged during the past several decades.

In recent years, the keyword "data" attracts people's attention due to the highly developed internet and communication techniques. That is the main reason why internet companies (Google, Amazon and many others) were trying to mine the values of the collected data underlying the internet. A trend is that machine learning revives, more precisely speaking "deep learning". One branch of the hot areas is the design of various neural networks(such as LSTM, CNN, LSTM-CNN), which form the basis of the deep learning techniques and are deemed as universal approximaters. It seems that the "art of fitting"(data modeling) is gradually approaching the "art of reasoning"(mathematical modeling) if dataset for training is large enough.

Engineering data are also valuable and widely accessible. How can engineering problems in data forms be investigated by deep learning. In general, there are two topics: 1) data <-> physical laws, 2) data <-> solution of BVPs. There have been tremendous researches on these two topics. It is questionable that the deep learning techniques can surpass mathematical reasoning and traditional numerical methods. The first question is how large the dataset should be? The second question is the portability of the trained nueral networks, that is to say, once a neural network is trained for a specific BVP, it can not be accommodated to another one. Thus, the deep learning techniques for discovery or solution still cease at a very limited level while the well-documented framework "Tensorflow" seems considerably powerful (e.g. graph calculation scheme and symbolic operators), which opens doors for researches in many different disciplines.

References can be found here:

  1. Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations

  2. Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations

  3. Distilling Free-Form Natural Laws from Experimental Data

An example about physics informed deep learning is shown as following

Caption: (a) A schematic showing a cantilever beam subject to a tip time-varying force. The displacement data is extracted from the 3 points marked by red cross. (b) showing the geometry parameters, material parameters and the force function. (c) the prediction compared with the exact one.

The source code and training dataset can be found here: script-> Beam and data-> Data

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