ML Adaptive Mesh Refinement

It's a method that uses ML to predict which regions should be refined based on error estimates or solution gradient patterns. It reduces the computational cost of a posteriori error estimation while enabling efficient h-refinement.

Expressing this mathematically, it looks like this.

ML prediction model:

ML adaptive mesh refinement is an important technique aiming to fuse data-driven approaches with physics-based modeling. While computational cost is a major bottleneck in traditional CAE analysis, introducing ML adaptive mesh refinement 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 spaces is crucial for practical application, and approaches like dimensionality reduction and leveraging sparsity are important.

It presents 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 must sufficiently represent the physics of the analysis target.
• The relationship between input parameters and output must be smooth (if discontinuities exist, domain partitioning is necessary).
• 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 vs. diffusion. Pe >> 1 indicates convection dominance (stabilization techniques 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 an ill-posed problem, 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.

Explains the numerical methods and algorithms for implementing ML adaptive mesh refinement.

As data preprocessing, normalization/standardization of input features is crucial. Since CAE data scales vary greatly for different physical quantities, appropriate selection of Min-Max normalization or Z-score normalization is necessary. For learning algorithm selection, the appropriate method should be chosen based on data volume, dimensionality, and degree of nonlinearity.

Implementation using the Python ecosystem (scikit-learn, PyTorch, TensorFlow) is common. Keys to implementation are learning acceleration via GPU parallelization, automatic hyperparameter tuning, and preventing overfitting through cross-validation. Using the HDF5 format is recommended for efficient I/O processing of large-scale CAE data.

It's important to use k-fold cross-validation, Leave-One-Out method, and holdout method appropriately for the purpose, and to evaluate prediction performance comprehensively using coefficient of determination R², RMSE, MAE, and maximum error.

Ensure code quality and experiment reproducibility by introducing version control (Git), automated testing (pytest), and CI/CD pipelines. Strictly enforce dependency version pinning (requirements.txt) to make rebuilding the computational environment easy. Ensuring result reproducibility by fixing random seeds is also an important implementation practice.