AI for Mechanical Engineering

Physics-informed Neural Networks

Problem 1ΒΆ

Cost function of PINN can be written as below. Describe what each terms in the cost function means when the given ODE system is $ {u^\prime} = f(u,t)$ with $t \in [0, 1]$ and a known initial condition $u(0) = u_0$. $ {\text{NN}(t)}$ is the solution by a neural network.

$${L(\omega)} = \sum_{i} \left(\frac{d \text{NN}(t_i)}{dt}-f(\text{NN}(t_i),t_i)\right)^2 + (\text{NN}(0)-u_0)^2 $$

Solution 1

  • $\sum_{i} \left(\frac{d \text{NN}(t_i)}{dt}-f(\text{NN}(t_i),t_i)\right)^2$ : ODE Loss

  • $(\text{NN}(0)-u_0)^2$ : Initial Condition Loss