Parameter identification by physics-guided neural network

The laws of physics enable us to predict observations when the parameters of a system are known. For example, knowing the geometrical characteristics and mechanical properties of the material(s) of a musical instrument, it is possible to predict the sound that will be emitted for a given excitation. This type of problem is known as "direct". Conversely, an "inverse" problem consists, for example, in predicting a geometrical characteristic of the instrument from observations, in this case recordings of the sound emitted. Without a priori knowledge, i.e. using only observations (such as sound), the problem is ill-posed in the sense that several geometries are possible. One of the crucial characteristics of an ill-posed problem in Hadamard's sense [F 1 380] is the non-uniqueness of the solution, which is the case for an inverse problem. In this case, the a priori helps to reduce the field of possibilities and can be integrated into the solution in the form of regularization terms, for example. More generally, an inverse problem involves determining the properties of a system from observations. This type of problem is encountered in many fields of engineering, when one wishes to estimate the properties of a medium inaccessible to direct observation, from observations made outside it .

Solving an inverse problem can involve identifying the parameters of a system from certain data. The method of parameter identification depends on the nature of the problem, particularly if the system is non-linear, and the challenge is to propose a model that explains the data and has predictive capabilities. One way of choosing the model is to calibrate it through a loss function optimization process, based for example on non-linear least squares, during which the deviation between the observed data and the calculated solution is evaluated and then the parameters modified so that the solution evolves in the direction that minimizes this deviation the fastest. This type of approach is at the heart of a new family of problem-solving methods, both direct and inverse, which take neural networks as their model. These new-generation neural networks incorporate in their loss function an original contribution linked to the laws of physics, generally described by differential equations. Using this principle, Raissi et al.