This paper presents an efficient mesh deformation method based on boundary integration and neural operators, formulating the problem as a linear elasticity boundary value problem (BVP). To overcome the high computational cost of traditional finite element methods and the limitations of existing neural operators in handling Dirichlet boundary conditions for vector fields, we introduce a direct boundary integral representation using a Dirichlet-type Green's tensor. This formulation expresses the internal displacement field solely as a function of boundary displacements, eliminating the need to solve for unknown tractions. Building on this, we design a Boundary-Integral-based Neural Operator (BINO) that learns the geometry- and material-aware Green's traction kernel. A key technical advantage of our framework is the mathematical decoupling of the physical integration process from the geometric representation via geometric descriptors. While this study primarily demonstrates robust generalization across diverse boundary conditions, the architecture inherently possesses potential for cross-geometry adaptation. Numerical experiments, including large deformations of flexible beams and rigid-body motions of NACA airfoils, confirm the model's high accuracy and strict adherence to the principles of linearity and superposition. The results demonstrate that the proposed framework ensures mesh quality and computational efficiency, providing a reliable new paradigm for parametric mesh generation and shape optimization in engineering.
Figures & tables
Figure 1: Deformation of Flexible beam : (a) initial mesh (1980 triangles). (b) deformed mesh (mesh topology and the number of triangles remain the same).
Figure 2: Element quality metrics of flexible beam distorted from the top and bottom layers of beam: (a) the deformed mesh generated by FEM. (b) the deformed mesh generated by the proposed surrogate-model.
Figure 3: The proposed surrogate model for the solution of Eq. ( 1 ) in this case : (a) the solution in X direction. (b) the solution in Y direction. Both evaluate the approximation accuracy by the point-wise squared error.
Figure 5: Element quality metrics of NACA 0012 airfoil under combined translational and rotational rigid-body motion: (a) deformed mesh generated by FEM. (b) deformed mesh generated by the proposed surrogate model.
Figure 6: The proposed surrogate model for the solution of Eq. ( 1 ) in this case : (a) the solution in X direction. (b) the solution in Y direction. Both evaluate the approximation accuracy by the point-wise squared error.
Figure 7: The verification of the scale invariance of the NACA 0012 airfoil example: (a) The deformed mesh used as the scale reference. (b) The quantitative relationship between the scale scaling factor and output response.
Figure 8: The verification of the scale invariance of the flexible beam example: (a) The deformed mesh used as the scale reference. (b) The quantitative relationship between the scale scaling factor and output response.
Scale k
RLE
Scale k
RLE
0.1
3.1335×10−7
1.2
3.3118×10−7
0.5
0.0000×100
1.5
3.2918×10−6
0.8
3.1335×10−7
2.0
0.0000×100
1.0
0.0000×100
2.5
3.3552×10−7
Table 1: Relative Linearization Error (RLE) for Different Scale Parameter Combinations in the NACA 0012 airfoil
Scale k
RLE
Scale k
RLE
0.1
1.5924×10−7
1.2
1.4415×10−7
0.5
0.0000×100
1.5
1.6150×10−6
0.8
1.5924×10−7
2.0
0.0000×100
1.0
0.0000×100
2.5
1.6491×10−7
Table 2: Relative Linearization Error (RLE) for Different Scale Parameter Combinations in the flexible beam
Figure 9: The verification of the superposition principle of the surrogate model: (a) is the deformed mesh generated by the NACA 0012 airfoil after rotation and translation superposition under the standard scale (a=1, b=1). (b) is the superposition error under the standard scale.
Combination
Parameter a
Parameter b
RSE
1
1.0
1.0
3.7663×10−6
2
2.0
0.5
4.4212×10−7
3
-1.0
1.0
3.0857×10−7
4
0.5
-1.5
3.2926×10−7
5
0.1
2.0
3.6614×10−7
6
2.5
2.5
3.7587×10−6
Table 3: Relative Superposition Error (RSE) for Different Superposition Parameter Combinations
Mesh-free numerical methods provide flexible discretisations for complex geometries; however, classical meshless discrete differential operators typically trade low computational cost for limited accuracy or high accuracy for substantial per-stencil computation. We introduce a parametrised framework for learning mesh-free discrete differential operators using a graph neural network trained via polynomial moment constraints derived from truncated Taylor expansions. The model maps local stencils relative positions directly to discrete operator weights. The current work demonstrates that neural networks can learn classical polynomial consistency while retaining robustness to irregular neighbourhood geometry. The learned operators depend only on local geometry, are resolution-agnostic, and can be reused across particle configurations and governing equations. We evaluate the framework using standard numerical analysis diagnostics, showing improved accuracy over Smoothed Particle Hydrodynamics, and a favourable accuracy-cost trade-off relative to a representative high-order consistent mesh-free method in the moderate-accuracy regime. Applicability is demonstrated by solving the weakly compressible Navier-Stokes equations using the learned operators.
Lucas Gerken Starepravo, Georgios Fourtakas, Steven Lind +3
School of Engineering, The University of Manchester, Manchester, UK · School of Engineering, Cardiff University, Cardiff, UK
Operator learning in scientific machine learning is concerned with approximation of maps between infinite-dimensional function spaces; such maps frequently arise as the solution operators of partial differential equations (PDEs). Neural operators have demonstrated broad empirical success at approximating such maps from data. However, most existing neural operator architectures enforce boundary conditions indirectly through training from data even though the boundary condition is often known exactly. Furthermore, existing modifications and approaches that do enforce boundary conditions explicitly suffer from impractical restrictions, including boundary smoothness, uniform grids, and separable, box-like domains. In this work, we propose an architecture which, independently of training, satisfies homogeneous Dirichlet boundary conditions, whilst simultaneously retaining the expressivity of existing kernel-integral neural operator architectures. This is achieved by enforcing the property that the output of each layer is contained in the span of a subset of the homogeneous Dirichlet eigenfunctions of the Laplacian on the output domain. The method requires only that the output domain be bounded with Lipschitz boundary and places no restriction on the choice of discretization, making it applicable to arbitrary mesh data and general geometries. We prove universal approximation for the resulting architecture; furthermore the approach we adopt in the analysis proves universality for a broad class of kernel-integral neural operators thereby uniting existing theory for a variety of operator learning methods. We validate the proposed method on maps defined by the coefficient to solution map in 2D PDEs: Darcy flow on a square domain and the Helmholtz equation on a circular domain. Comparisons are made with alternative methods.
Neural operators have emerged as powerful data-driven solvers for PDEs, offering substantial acceleration over classical numerical methods. However, existing transformer-based operators still face critical challenges when modeling PDEs on complex geometries: directly processing over massive mesh points is computationally expensive, while operating in raw discretization coordinates may obscure the intrinsic geometry where physical interactions are more naturally expressed. To address these limitations, we introduce the Charted Axial Transformer Operator (CATO), a geometry-adaptive and derivative-aware neural operator for PDEs on general geometries. Instead of applying attention directly in the physical coordinate system, CATO learns a continuous latent chart that maps mesh coordinates into a learned chart space, where chart-conditioned axial attention efficiently captures long-range dependencies with reduced computational cost. In addition, CATO introduces a derivative-aware physics loss for steady-state PDEs that jointly supervises solution values, mesh-consistent gradients, and an auxiliary flux-like field, improving physical fidelity and reducing oversmoothing. We further provide a theoretical approximation result showing that, under a favorable chart, charted axial attention can represent low-rank axial solution operators with controlled error, and that small chart perturbations induce bounded approximation degradation. CATO achieves the best performance across all evaluated datasets, yielding an average improvement of approximately 26.76% over the strongest competing baselines while reducing the number of parameters by 81.98%. These results highlight the effectiveness of learning geometry-adaptive charts and derivative-aware physical supervision for accurate and efficient PDE operator learning.