 |
Consider a general form of partial differential equation as an example, assuming that the function u = u(t, x) and satisfies a partial differential equation in the following form:
where N[u;ル] represents a parameterized non-linear operator that operates on u with parameter ル. ヘ denotes the Euclidean geometric space, and T is the terminal time. Given the initial state, boundary conditions, and physical parameter ル for u(t, x), solving the partial differential equation allows us to predict the values of u(t, x) at any spatiotemporal point. Numerical methods such as finite differences or finite elements can be employed for solving. First, let's define:
Subsequently, a neural network can be constructed to approximate the solution to the partial differential equation. Therefore, the root mean square loss function is defined as following:
the data-driven of the loss function, acquired through the initial and boundary condition in the training data, represents the physics model-driven portion of the loss function, which is trained in the system's machine learning process to approximate the neural network function. The network must not only minimize the error with known training data but also satisfy the constraints of the partial differential equation as much as possible. Therefore, the network structure and loss function of PINN need to be customized according to the form of the partial differential equation, while also maintaining the model's scalability.
PINNs - Neural networks that are trained to solve supervised learning tasks while respecting physical laws. Many physical systems consist of prior knowledge encoded in physical laws, e.g. Newton¨s laws of motion. This prior information can act as a regularization constraint that reduces the space of admissible solutions , remove unrealistic solutions that violate fundamental conservation laws mass, momentum, energy
|
IEEE/ICACT20240208 Slide.04
[Big Slide]
[YouTube]