DeepONet Operator Learning

DeepONet is an architecture that learns function-to-function mappings (operators). It encodes the input function with the Branch Net, processes the output location with the Trunk Net, and combines them.

Expressing this mathematically looks like this.

Loss function:

DeepONet operator learning 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 DeepONet operator learning 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 theoretical research topics including guarantees of generalization performance and rigorous analysis of convergence. Particularly, dealing with the "curse of dimensionality" when the input dimension is high is a key practical challenge, and approaches like dimensionality reduction and leveraging sparsity are important.

It 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 equations, 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 (if discontinuities exist, domain decomposition is needed).
• Reducing computational cost is the main purpose; 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.

• Péclet Number Pe: Relative importance of convection vs. diffusion. Pe >> 1 indicates convection-dominated (stabilization methods required).
• 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.

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 getting your hands dirty is the best way to learn. If you have any questions, feel free to ask anytime.