# Where to apply Physics Based Machine Learning?

Physics based Machine Learning is growing fast by merging both the machine learning & the numerical methods techniques. However PBML is outperformed by traditional approaches in case of well-posed forward problems.

The well-posed problems are well defined boundary & initial conditions. A system of partial differential equations is well-posed if

it has a solution &

the solution is unique, &

the solution depends continuously on the initial values.

In many cases it is difficult to decide if a system is well-posed or not. Often the answer is not known.

The forward problem is a particular model that calculates what should be observed. Traditional solvers are sufficient to address these types of problems.

On the other hand, there are only data available without any interesting physics (or modelling) behind it. In this case, traditional machine learning approaches are sufficient to obtain insights of the models.

Then why PBML?

1. The need for physics informed learning comes only when models are incomplete & data are scattered noisy. As PINN inherently smooths data, meaningful solutions are possible with problems that are not well-posed.

The (forward/inverse) problems, without any initial or boundary conditions or lacking parameters in the PDEs, are not possible to solve with regular methods.

2. Next attractive point is the regime with small data. Unlike the traditional machine learning techniques that require volume of data, PBML has the strong generalisation in the small data regime itself. The physics pays the penalty by augmenting data by going with ML.

3. It can also tackle higher dimensionality associated with input space. Deep Operator Network (DeepONets) was demonstrated succesfully with operator regression & applications to PDEs.

4. Uncertainty quantification is a key factor in forecasting the evolution of systems. While the source of uncertainty springs from physics, data & the learning model… the physics informed GANs, along with tackling the curse of dimensionality, handles stochastic dimensionality effectively to solve physics based uncertainty.

The 2nd uncertainty is associated with gaps or noise in the data. Bayesian approach manages to solve them. However, setting prior is still an open question.

The 3rd uncertainty is limitation of learning capability & convolutional encoder-decoder NN maps the source term to tackle them.

Thus these 4 scenarios compel one to for Physics Based Machine Learning for analysis.

Now quantum algorithms are waiting to pave ways for more effective solutions in & with PBML. Fingers crossed.

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Image: go.nature.com/2ZweNuG