Local Mesh Assembly Training

• Local Mesh Assembly Training is a framework that uses machine learning to drive adaptive mesh refinement based on localized error estimates and geometric features.
• It leverages neural network architectures, such as feedforward and residual networks, to predict optimized local mesh densities for efficient finite element analysis.
• The approach integrates precise error norms and adaptive refinement mechanisms to reduce computational cost while achieving higher simulation accuracy compared to traditional methods.

Local mesh assembly training refers to data-driven and machine learning frameworks that enable the construction or refinement of mesh elements or subdomains based primarily on locally inferred, predicted, or optimized features. In computational geometry, finite element analysis, and computer vision, the assembly of high-quality meshes with locally adapted resolution, topology, and connectivity is critical for accurate simulations or scene understanding. Recent approaches integrate neural networks, reinforcement learning, and statistical inference to automate and optimize the local assembly of meshes, replacing or complementing traditional heuristic- or error-driven manual design. This entry provides a comprehensive, technical overview of methodologies, architectures, mathematical underpinnings, and application scenarios for local mesh assembly training across computational science, graphics, and geometry processing.

1. Supervised Learning of Local Mesh Density

Supervised local mesh assembly training regimes focus on learning mappings from geometric, physical, and parametric inputs to local mesh refinement targets—typically mesh density or sizing fields. In "MeshingNet: A New Mesh Generation Method based on Deep Learning" (Zhang et al., 2020), local mesh density guidance is learned by supervising a deep neural network (MeshingNet) to predict the optimal local element area $A(x)$ for finite element (FE) problems. The training set is constructed offline using a two-mesh error estimation workflow: high-density (HDUM) and low-density (LDUM) uniform meshes are used to solve the underlying PDEs, and an a posteriori error estimator computes element-wise errors $E(x_i)$ by comparing the solutions (after interpolation).

The network receives as input both global (geometry $\Gamma$, boundary condition $B$, material parameter $M$) and local (spatial location $X$) features, and is trained to perform the regression: $F: (\Gamma, B, M, X) \to A(X)$ where $A(x_i) = K / E(x_i)^\alpha$ with $K$ as a domain-dependent scaling constant and $\alpha$ (typically $1$) controlling the error-adaptivity. The loss is mean squared error, optimized with Adam and a batch size of 128. Training can leverage millions of LDUM element samples, producing highly generalized models. The resulting network can rapidly infer a domain-conforming mesh density field, which, when consumed by standard mesh generators (such as Triangle), yields refined non-uniform meshes exhibiting superior error characteristics compared to uniformly refined or heuristic-adapted meshes.

2. Neural Architectures and Topologies

Architectural choices play a critical role in local mesh assembly training efficacy and convergence. In MeshingNet, both standard feedforward fully connected networks and multiple forms of residual networks (ResNets) are evaluated. Residual connections, either between early and late hidden layers or across all layers as in DenseNet-style architectures, significantly improve convergence and stability, particularly in learning the highly nonlinear mappings from geometric and boundary condition inputs to mesh density. All hidden layers employ ReLU activations, and input dimensionality is problem-specific but flows through a canonical progression: input $\rightarrow$ 32 $\rightarrow$ 64 $\rightarrow$ 128 $\rightarrow$ 128 $\rightarrow$ 64 $\rightarrow$ 32 $\rightarrow$ 8 $\rightarrow$ 1.

During inference, the trained model outputs local density bounds for all relevant spatial locations, which enables immediate generation of the mesh density guide. Batch inference times per problem are on the order of 0.046 seconds, enabling real-time deployment in simulation workflows.

3. Error Norms and Adaptivity Mechanisms

Error equidistribution and adaptive refinement are central to local mesh assembly. The assembly network is trained to predict local mesh sizes inversely proportional to estimated FE solution errors, i.e., elements with higher estimated error receive finer (smaller) area bounds. The error $E(x)$ can be any user-defined norm: for Poisson’s equation, an $L_1$ relative error; for linear elasticity, an energy norm $$E = (\varepsilon^L - \varepsilon^H) : (\sigma^L - \sigma^H)$$. Designating the norm is thus a point of application-specific flexibility.

Explicit formulae anchor the methodology, ensuring that the learning process can flexibly adapt to different application requirements. As shown, the predicted field is proportional to the inverse of the local a posteriori error,

$A(x_i) = K / E(x_i)^\alpha.$

This results in spatially variable, high-resolution meshes only where they are strictly necessary for accuracy, thus reducing overall element counts and computational burden.

4. Performance Characterization and Evaluation

Empirical evaluation is typically conducted on canonical PDE problems:

• For a 2D Poisson problem on random octagonal domains, MeshingNet-generated meshes decrease relative $L_1$ errors from $[0.0015, 0.0025]$ (uniform mesh with 4000 elements) to $[0.0007, 0.0015]$.
• For linear elasticity with plane stress and heterogeneous boundary conditions, the potential energy of solutions is closer to that of the high-density mesh ground truth (e.g., mean energy $-7.681$ on predicted meshes vs. $-7.729$ for the high-accuracy mesh).

In both cases, non-uniform, locally adapted meshes significantly outperform both uniform and standard heuristic-refined meshes on the size-accuracy trade-off. Once trained, the model’s prediction time is orders of magnitude lower than traditional iterative adaptive methods.

5. Mathematical Formulation and Theoretical Basis

Local mesh assembly training relies on clear mathematical formalism:

• Data mapping:

$F: (\Gamma, B, M, X) \mapsto A(X)$

• Area upper bound prediction:

$A(x_i) = K / E(x_i)^\alpha$

• Error estimators:

$$E = \left| \frac{u^{\text{LAS}} - u^{\text{HAS}}}{u^{\text{HAS}}} \right| \quad \text{(Poisson)}$$

$$E = (\varepsilon^L - \varepsilon^H) : (\sigma^L - \sigma^H) \quad \text{(Elasticity)}$$

• FE energy functional:

$$E_p = \int \left( \frac{1}{2} \varepsilon : \sigma - F \cdot u \right) d\Omega - \int (\sigma \cdot u) d\Gamma$$

These equations ensure both implementation clarity and analytic control over the training regime.

6. Application Scenarios and Generalization

This approach is agnostic to geometry (arbitrary polygons), boundary conditions (fixed, traction, mixed), and PDE parameters, due to the inclusion of all these factors as network inputs. The method is demonstrated for both single PDEs and coupled systems. The flexibility in error norm selection permits application across a range of engineering and scientific domains where mesh adaptivity directly improves computational tractability without sacrificing solution accuracy.

An important practical consideration is that the learned model is not restricted to the geometries seen during training: the inclusion of a large, diverse set of domain samples during offline generation enables robust generalization. Once trained, for any new FE problem (geometry, material, or boundary condition), the model can rapidly output a suitable mesh density field with negligible inference cost.

7. Impact and Technical Significance

Local mesh assembly training, as exemplified by MeshingNet, replaces costly iterative or manual mesh refinement with rapid, learned prediction. This methodology has measurable technical advantages:

• Orders of magnitude speedup in mesh adaptivity phases of simulation workflows.
• Flexibility to user-chosen error norms and quality criteria.
• Capacity to exploit detailed geometric, material, and boundary information, with generalization to unseen configurations.
• Compatibility with existing FE solvers and mesh generators by providing density guides in standard formats.

By explicitly encoding the data generation, error estimation, and regression workflow, local mesh assembly training positions itself as a foundational approach for scalable, accurate, and automated mesh generation in computational science and engineering applications (Zhang et al., 2020).