Physics-Informed Neural Networks (PINN) Fundamentals

Physics-Informed Neural Networks (PINN) are a data-driven method that learns solutions satisfying physical laws by incorporating governing equations into the neural network's loss function.

This can be expressed mathematically like this.

PDE Residual Loss:

The foundational theory of PINN is an important methodology aiming to fuse data-driven approaches and physics-based modeling. While computational cost is a major bottleneck in conventional CAE analysis, introducing the foundational theory of PINN can significantly improve the trade-off between computational efficiency and prediction accuracy. The mathematical foundation of this method is based on function approximation theory and statistical learning theory, with generalization performance guarantees and rigorous analysis of convergence being key theoretical research topics. Particularly, dealing with the "curse of dimensionality" in high-dimensional input cases is crucial for practical application, and approaches like dimensionality reduction and leveraging sparsity are important.

It shows the basic mathematical framework for applying machine learning models to CAE.

The loss function in AI×CAE is composed as a weighted sum of data-driven terms and physics constraint terms:

Here, $\mathcal{L}_{\text{data}}$ is the squared error with observed data, $\mathcal{L}_{\text{physics}}$ is the residual of the governing equation, and $\mathcal{L}_{\text{reg}}$ is a regularization term. Adjusting the weight parameters $\lambda$ greatly affects learning stability and accuracy.

The biggest challenge for surrogate models is prediction accuracy outside the range of training data (extrapolation region). Incorporating physical laws can improve extrapolation performance, but complete guarantees are difficult.

When the dimension of the input parameter space is high, the required number of samples increases exponentially. Efficient sample placement through Active Learning or Latin Hypercube Sampling (LHS) is super important.

• The training data sufficiently represents the physics of the analysis target.
• The relationship between input parameters and output is smooth (domain decomposition is needed if there are discontinuities).
• Reducing computational cost is the main objective; conventional solvers should be used in conjunction for final verification requiring high accuracy.
• If the quality of training data (mesh-converged, V&V completed) is insufficient, model reliability decreases.

Understanding the dimensionless parameters governing the physical phenomenon being analyzed is the foundation for appropriate model selection and parameter setting.

• Peclet Number Pe: Relative importance of convection and diffusion. Pe >> 1 indicates convection-dominated (stabilization techniques are needed).
• Reynolds Number Re: Ratio of inertial forces to viscous forces. A fundamental parameter for fluid problems.
• Biot Number Bi: Ratio of internal conduction to surface convection. For Bi < 0.1, the lumped capacitance method is applicable.
• Courant Number CFL: Indicator of numerical stability. For explicit methods, CFL ≤ 1 is required.

For order-of-magnitude estimation of analysis results, dimensional analysis based on Buckingham's Π theorem is effective. Using characteristic length $L$, characteristic velocity $U$, and characteristic time $T = L/U$, the order of each physical quantity is estimated beforehand to confirm the validity of the analysis results.

Selecting appropriate boundary conditions is directly linked to solution uniqueness and physical validity. Insufficient boundary conditions lead to ill-posed problems, while excessive ones cause contradictions.

Yeah, you're doing great! Actually getting hands-on is the best way to learn. If you have any questions, feel free to ask anytime.