Solution of partial differential equations using neural networks
The solution of PDEs is of critical importance to a large variety of physical problems. However, except for the most elementary cases, there are no analytical closed form solution to most PDEs describing phenomena in practical applications. There are various numerical ways that can be used to solve these equations, some more computationally efficient, and accurate than others. In my own research I have spent a significant amount of time implementing finite difference schemes to solve partial differential equations. This can be very computationally intensive and can take very long, even on multi-core computing clusters.
One alternative is to use artificial neural networks to solve the PDEs. The idea has been explored in some detail in the existing literature (for example see here). The key insight here is that we use a neural network as a function approximator. We write the solution as the sum of two terms; the first term satisfies the boundary conditions, while the second term is constructed in such a way as to not contribute to the boundaries. The neural network is then trained over the domain of the equation by minimizing the squared loss when the equation is written in the form LHS=0 (i.e.) we force the neural network to adjust it’s parameters so that the constructed solution satisfies the equation. More details and notes on the implementation can be found in the notebook.