Multi-Layer Perceptron
Physics-Informed Neural Networks (PINNs)
DeepONet: Operator Learning with Neural Networks
Differentiable Simulation
Graph Neural Network
Lab Setup

Stage 2: The PINN Approach to Inverse Problems

The Key Insight:

Traditional inverse methods:

  • Iterate between parameter guessing and forward solving
  • Computationally expensive
  • Prone to local minima

PINN approach:

  • Treat unknown parameters as additional trainable variables
  • Optimize network weights AND physical parameters simultaneously
  • Use physics as regularization

The PINN Inverse Framework

For our heat equation problem, we have:

Unknowns to learn:

  1. \( \hat{T}_\theta(x) \) - neural network approximating temperature
  2. \( \hat{\kappa} \) - estimated thermal diffusivity (scalar parameter)

Loss function components:

\[ \mathcal{L}_{\text{total}}(\theta, \hat{\kappa}) = w_1\mathcal{L}_{\text{data}} + w_2\mathcal{L}_{\text{PDE}} + w_3\mathcal{L}_{\text{BC}} \]

where:

Data Loss: \( \mathcal{L}_{\text{data}} = \frac{1}{N_d}\sum_{i=1}^{N_d}|\hat{T}_\theta(x_i) - T_i|^2 \)

Physics Loss: \( \mathcal{L}_{\text{PDE}} = \frac{1}{N_f}\sum_{j=1}^{N_f}\left|-\hat{\kappa}\frac{d^2\hat{T}_\theta}{dx^2}(x_j) - f(x_j)\right|^2 \)

Boundary Loss: \( \mathcal{L}_{\text{BC}} = |\hat{T}_\theta(0) - T_0|^2 + |\hat{T}_\theta(1) - T_1|^2 \)

Why This Works

  1. Simultaneous optimization: Both \( \theta \) and \( \hat{\kappa} \) are updated together
  2. Physics regularization: The PDE constraint guides parameter estimation
  3. Data efficiency: Physics fills gaps between sparse measurements
  4. Robustness: Physics constraints filter noise in data
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CVW material development is supported by NSF OAC awards 1854828, 2321040, 2323116 (UT Austin) and 2005506 (Indiana University)