Fourier Neural Operator (FNO)

An architecture that efficiently learns PDE solution operators through convolution in Fourier space. Its characteristic is resolution-independent generalization performance.

Expressing this mathematically, it looks like this.

Fourier layer kernel integral:

The Fourier Neural Operator (FNO) is an important method aiming to fuse data-driven approaches and physics-based modeling. While computational cost is a major bottleneck in conventional CAE analysis, introducing the Fourier Neural Operator (FNO) 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 rigorous analysis of generalization guarantees and convergence being key theoretical research topics. Particularly, dealing with the "curse of dimensionality" when the input dimension is high is crucial for practical application, and approaches like dimensionality reduction and leveraging sparsity are important.

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

In AI×CAE, the loss function is composed as a weighted sum of a data-driven term and a physics constraint term:

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 extremely 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 discontinuities exist).
• 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 forms the basis for appropriate model selection and parameter setting.

• Péclet Number Pe: Relative importance of convection and diffusion. Pe >> 1 indicates convection-dominated (stabilization methods required).
• Reynolds Number Re: Ratio of inertial forces to viscous forces. Fundamental parameter for fluid problems.
• Biot Number Bi: Ratio of internal conduction to surface convection. Lumped capacitance method applicable when Bi < 0.1.
• 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.

Choosing 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 trying things out is the best way to learn. If you don't understand something, feel free to ask anytime.