DeepONet-Transolver: Hybrid Operator Learning

• DeepONet-Transolver is a hybrid framework that combines deep operator networks with attention-based transformers to deliver simultaneous spatial and temporal predictions.
• It employs mesh-based attention and latent operator learning to accurately predict static nodal displacements and dynamic reaction force histories in PET bottle buckling analysis.
• The approach achieves low relative L2 errors and high R2 correlations, enabling rapid, scalable multi-physics simulations that outperform conventional finite element analysis.

The DeepONet-Transolver framework refers to the integration of deep operator networks (DeepONet) with physics-informed attention-based transformer solvers (Transolver) to address the challenges of predicting complex physical responses—specifically for nonlinear, nonparametric geometric domains. This hybrid approach has been developed for applications such as PET bottle buckling analysis, providing accurate and rapid multi-task predictions (static nodal fields and dynamic force histories) that are computationally prohibitive with conventional finite element analysis (FEA) (Kumar et al., 16 Sep 2025).

1. Hybrid Framework Structure and Design

The DeepONet-Transolver architecture is structured to exploit the complementary strengths of both DeepONet and Transolver:

• Transolver Component: Handles complex, non-Euclidean geometry using an attention-based mechanism. The local mesh geometry, represented as a point cloud with node coordinates and surface normals, is projected via an MLP into a high-dimensional latent space and reorganized into "slices" corresponding to multiple attention heads. A softmax operation produces weights reflecting physical similarity, enabling aggregation within and across physically consistent regions. The resulting physics-aware tokens are then further processed and desliced (dispersed back to mesh nodes) to produce the spatial prediction—here, the final nodal displacement fields for each PET bottle geometry.
• DeepONet Component: Models time-dependent (e.g., transient or evolutionary) operator learning for observables such as reaction force histories. The branch network ingests the geometry-specific latent representation from Transolver (aggregated by pooling). The trunk network receives the time coordinate $t$ as input. The DeepONet output, representing the functional mapping from geometry and time to the observable (e.g., top-load reaction force), is computed as

$$H_\theta: \mathcal{K}(\Omega_i) \times T \to \mathbb{R}$$

where $\mathcal{K}(\Omega_i)$ denotes the geometry’s latent encoding and $T$ denotes the time domain.

The integrated workflow allows the model to simultaneously output static fields (nodal displacements) and time-evolving quantities (reaction forces), respectively corresponding to spatial and temporal aspects of the bottle compression physics.

2. Application to Non-Parametric Bottle Buckling

The framework targets the buckling analysis of PET bottles under quasi-static axial compression—a representative multi-physics design problem in packaging:

• Input: Detailed 3D meshes of bottles, parameterized by either 2 or 4 design parameters, comprising ~20,000 nodes each. For every geometry, time-dependent FEA simulation in Abaqus provides the nodal displacements at the end of loading and the full reaction force–displacement (load–deflection) curve as a function of compression distance.
• Learning Task: Learn the operator

$$\mathcal{G}: \{\Omega_i\} \to u_i(\mathbf{x}), \qquad \mathbf{x} \in \Omega_i \subset \mathbb{R}^3$$

and the time-dependent operator for reaction forces,

$$F_R(t) = \mathcal{H}_\theta(\text{latent}(\Omega_i), t)$$

where $u_i(\mathbf{x})$ is the displacement field and $F_R(t)$ the reaction force at time $t$.

• Transolver enables geometry-aware learning by discovering consistent local behaviors across irregularly meshed surfaces (e.g., ribs vs. smooth regions), enhancing generality to nonparametric shape variations.
• DeepONet enables operator learning over the latent representation and time, so that the full reaction force trajectory can be inferred as a function of both the bottle shape and simulation progress.

3. Training Protocol and Evaluation Strategy

• Data Generation: For each of 254 unique geometries per design family (2- or 4-parameter), FEA yields the target responses. Mesh node features include coordinates and surface normals; the ground truth includes $u_x$, $u_y$, $u_z$ at every node and $F_R(t)$ at discrete time steps.
• Transolver Training: The full mesh $\mathbf{g} \in \mathbb{R}^{N \times C}$ is passed through the attention-based architecture to output the $u(\mathbf{x})$ vector field at simulation completion.
• DeepONet Training: The pooled latent vector (from Transolver) is input to the branch net; the trunk net takes normalized time $t$. Model output is the predicted $F_R(t)$. The loss function is the sum of $L^1$ losses over all three displacement components and the reaction force.
• Evaluation Metrics: Relative $L^2$ norm is computed for both displacement and reaction force predictions; pointwise absolute errors and $R^2$ correlation coefficients between predicted and reference solutions are also analyzed.

4. Quantitative Performance and Physical Fidelity

• Mean relative $L^2$ error for displacement fields lies between $2.5\%$ and $13\%$ (higher for higher-dimensional design parameterization due to dataset sparsity).
• Mean relative $L^2$ error for reaction force curves is approximately $2.4\%$.
• Pointwise errors in nodal displacements are typically in the range $10^{-4}$–$10^{-3}$, with maxima in the neck and rib regions (high-gradient buckling zones).
• $R^2$ values for predicted vs. reference curves routinely exceed $0.98$–$0.99$.
• The framework robustly captures both the global deformation pattern and localized buckling events, accurately reproducing force drops and sudden deformation "kinks" associated with structural instability.

5. Technical Formulation and Key Equations

Table: Summary of Core Operators

Module	Input	Output	Key Operation
Transolver	Mesh features $\mathbf{g}$	$u(\mathbf{x})$	Attention-based tokenization, deslicing, MLP
DeepONet	Latent (pooled), $t$	$F_R(t)$	weighted sum over branch/trunk outputs

Main equations:

• Spatial mapping: $\mathcal{G}: \{\Omega_i\} \to u_i(\mathbf{x})$
• Latent aggregation: $\mathbf{y}^* = \text{MLP}(\text{tokens})$; pooling for DeepONet input
• Reaction force prediction: $$\hat{F}_R(t) = \sum_{i=1}^p b_i(\text{latent}) \cdot \text{tr}_i(t)$$
• Loss: $$\mathcal{L}_{\text{total}} = \mathcal{L}_{u_x} + \mathcal{L}_{u_y} + \mathcal{L}_{u_z} + \mathcal{L}_{F_R}$$

6. Scalability, Advantages, and Multitask Learning

• The hybrid framework is inherently multi-task: it simultaneously predicts the entire final displacement field (spatial) and the full reaction force time history (temporal) in a single forward pass.
• Mesh-based attention in Transolver allows adaptation to irregular geometries, making the approach suitable for families of shapes outside traditional parametric forms.
• Operator learning via DeepONet enables rapid prediction for unseen designs without needing to rerun costly FEA.
• Once trained, the surrogate operates orders of magnitude faster than direct simulation and is readily extensible to higher-dimensional design spaces as required by industrial applications.
• The slice-based attention in Transolver provides interpretability, implicitly identifying physically coherent regions (such as ribs or necks) and making the embedding robust to mesh perturbations or local geometric changes.

7. Significance and Future Outlook

The DeepONet-Transolver framework demonstrates that neural operator-based surrogates can generalize to physically significant nonlinear responses in nonparametric, complex domains:

• Effective coupling of geometry-aware attention mechanisms with operator learning yields highly accurate, scalable surrogates for both static (field) and dynamic (trajectory) responses in engineering design tasks.
• Provides a pathway for rapid multi-physics evaluation in settings that require repeated design analysis, optimization, or uncertainty quantification, without incurring the computational costs associated with FEA-based workflows.
• The framework’s combination of accuracy, scalability, and multitask capability suggests applicability to a broad class of computational mechanics problems beyond bottle buckling, contingent on further extensions to more complex or higher-dimensional geometries.

These findings establish a foundation for future development in physics-informed machine learning surrogates for design, with the ability to robustly handle arbitrary geometries and full-field dynamic responses through hybrid attention and operator learning (Kumar et al., 16 Sep 2025).