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What's the relationship of machine learning and mechanical simulation?

Particularly, machine learning is about learning from a large sample and predicting based on filters tuned on that.

Mechanical simulation is about using physical laws and seeing what numbers are produced using those.

If one deep learns mechanics, then what is this and what kind of properties does it have?

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Your question is a bit broad, I think. Lacking concreteness, I would give a high-level overview. Terms that you can search for are in italics.


Machine learning is a huge field, that includes many techniques. It has traditionally been used (and still is) in image processing, speech recognition, etc. In these applications, data are abundant.


What you call mechanical simulation belongs to the field of computational mechanics, and it is traditionally based on physical models. This approach is also called modelling from first principles.


What is the motivation to apply machine learning techniques in computational mechanics, or in computational science in general?

  1. Function approximation is problematic is high dimensions because of the huge growth of complexity (curse of dimensionality).
  2. Sometimes the mathematical model is not known. In mechanics, we got used to making simplifications to derive equations based on physical considerations (conservation laws, minimization of energy, symmetries, etc.). For complex, possibly interacting, systems such models are not available.
  3. Uncertainty in the parameters. Even if a mathematical model can be derived, it often contains parameters (e.g. coefficients of a PDE) whose values are not known.
  4. Whenever a parameter changes, the whole simulation has to be rerun again. This may take a lot of time.

Machine learning techniques can help to cope with these issues.

  1. Neural networks can be used for function approximation.
  2. Purely from data, using sparse identification, terms of a PDE or ODE can be identified. Many existing differential equations (e.g. Navier-Stokes) have been reinvented based on data (measurements) only. This way, new models can be generated.
  3. If the model is known but parameters are not, only can solve the so-called inverse problem to find the parameters that give rise to a specific result. Inverse methods existed long before scientific machine learning, but machine learning has proved to be a very successful way to solve these types of problems.
  4. Once the weights and biases of a neural network have been computed (i.e. the neural network has been trained), one can feed new input data, and the simulation is replaced by an interpolation, allowing orders of magnitude faster acquisition.

If one deep learns mechanics, then what is this

If deep learning is done by a deep neural network, it means that the parameters (weights and biases of each neuron) are determined. Next time it sees a similar problem, it can create the output quickly. Let the weights $w_i$ and the biases $b_i$ be arranged in a vector $\mathbf{\theta}$, which represents all the parameters of the network. In the training phase, the network is provided a solution $\mathbf{u}_0$ to the possibly nonlinear problem $\mathcal{N}(\mathbf{u}, \mathbf{p}) = 0$. Training the network means that its parameters $\theta$ are computed such that the solution operator $T:\ T\mathbf{p} = \mathbf{u}_0$ is learned. Hence, when new data $\tilde{\mathbf{p}}$ arrives, the already learnt operator can be applied on it to obtain the new solution: $\tilde{\mathbf{u}} = T\tilde{\mathbf{p}}$.

and what kind of properties does it have?

A properly trained neural network provides interpretability and generalizability. A scientific machine learning model is

  • interpretable if the generated solution satisfies the physical laws.
  • generalizable if it can accurately predict the solution ($\tilde{\mathbf{u}}$) when it meets a new, not yet know problem ($\tilde{\mathbf{p}}$).
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  • $\begingroup$ Very nice answer ! As someone that works in this area I just wanted to add that one of the main practical motivitations for using machine learning in mechanical simulations is that once the surrogate model is trained, obtaining a answers is instantaneous. $\endgroup$
    – coyote
    Nov 19, 2021 at 10:00
  • $\begingroup$ @coyote Thanks for reassuring me that what I wrote makes sense. I started working on this topic only a few weeks ago. $\endgroup$ Nov 19, 2021 at 16:19

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