Pde

There is a beautiful connection between probability theory and partial differential equations (PDEs), given by the famous Feynman–Kac theorem. This result allows us to translate between finite-dimensional stochastic problems and infinite-dimensional deterministic problems. It appears throughout finance, physics, control theory, and machine learning (generative modeling and reinforcement learning). What makes this connection especially useful is that it works in both directions. The solution to a high-dimensional PDE (which can be very expensive to compute numerically) can be evaluated at a single point simply by simulating a random process. Conversely, a difficult question about a stochastic process can be transformed into a deterministic PDE and tackled using standard techniques. ...

Classical data-driven methods in machine learning tend to be data-inefficient and they discard expert modeling knowledge. However, during the last decade a different approach has emerged under the name physics-informed machine learning. First posted in 2017 and published in 2019, physics-informed neural networks (PINNs) [6] combined deep learning with prior knowledge of the governing equations by directly learning the solution using the PDE residual as a soft penalty in the loss. In 2018, Neural ODEs [1] took a complementary route of embedding the physics in the model itself: the right-hand side of an ordinary differential equation $\dot y=f(y,t)$ is replaced by a neural network $f\approx f_\theta$, trained to reproduce observed data. The same principle was soon extended to stochastic differential equations [7] and to partial differential equations [5, 11, 12] — the latter is what we call here a Neural PDE. ...