3D Neural Operator-Based Flow Surrogates around 3D geometries: Signed Distance Functions and Derivative Constraints

TL;DR

This work addresses the computational bottleneck of 3D CFD by benchmarking DeepONet and Geometric-DeepONet as neural-operator surrogates for steady 3D flows around complex geometries encoded with Signed Distance Functions. The authors introduce derivative-informed loss functions (L1–L4) and a two-stage Geometric-DeepONet architecture to improve boundary-layer fidelity and velocity-gradient accuracy, evaluating on a FlowBench 3D lid-driven cavity dataset with $1{,}000$ samples spanning Reynolds numbers in $[10,1000]$ at $128^3$ resolution. Key findings show that Geometric-DeepONet with derivative-based losses delivers up to $32\%$ better boundary-layer accuracy and substantially enhances gradient accuracy (roughly $25\%$ in interpolation and up to $45\%$ in extrapolation), indicating improved generalization to unseen flow conditions. The results underscore the importance of geometry-aware representations and physics-informed training for reliable 3D flow surrogates, with practical implications for rapid design optimization and real-time analysis in engineering and biomedical contexts.

Abstract

Accurate modeling of fluid dynamics around complex geometries is critical for applications such as aerodynamic optimization and biomedical device design. While advancements in numerical methods and high-performance computing have improved simulation capabilities, the computational cost of high-fidelity 3D flow simulations remains a significant challenge. Scientific machine learning (SciML) offers an efficient alternative, enabling rapid and reliable flow predictions. In this study, we evaluate Deep Operator Networks (DeepONet) and Geometric-DeepONet, a variant that incorporates geometry information via signed distance functions (SDFs), on steady-state 3D flow over complex objects. Our dataset consists of 1,000 high-fidelity simulations spanning Reynolds numbers from 10 to 1,000, enabling comprehensive training and evaluation across a range of flow regimes. To assess model generalization, we test our models on a random and extrapolatory train-test splitting. Additionally, we explore a derivative-informed training strategy that augments standard loss functions with velocity gradient penalties and incompressibility constraints, improving physics consistency in 3D flow prediction. Our results show that Geometric-DeepONet improves boundary-layer accuracy by up to 32% compared to standard DeepONet. Moreover, incorporating derivative constraints enhances gradient accuracy by 25% in interpolation tasks and up to 45% in extrapolatory test scenarios, suggesting significant improvement in generalization capabilities to unseen 3D Reynolds numbers.

Figures (16)

• Figure 1: Visualization of flow streamlines for representative geometries from the CFD dataset at various Reynolds numbers. Panels (a) and (b) show the flow around a cube at $Re=13$ and $Re=933$, respectively; panels (c) and (d) show the flow around a torus at $Re=16$ and $Re=909$, respectively.
• Figure 2: Level-set visualization of the Signed Distance Field (SDF) for four representative geometries from our dataset, arranged in a 2$\times$2 layout. Panel (a) depicts the SDF of a cylinder, panel (b) shows the SDF of a cube, panel (c) illustrates the SDF of a ring, and panel (d) presents the SDF of an ellipsoid. The level-sets are displayed for three isocontours: SDF = 0 at the geometry's surface, SDF = 0.15, and SDF = 0.3 for two additional distances from the surface respectively.
• Figure 3: Comparison of DeepONet (top) and Geometric-DeepONet (bottom) on the 3D driven-cavity flow problem. DeepONetlu2020 uses a dual-network structure: the branch network (MLP) encodes input parameters, while the trunk network (MLP) processes spatial coordinates $(x,y,z)$. The two latent representations are fused via a dot product to predict velocity. Geometric-DeepONethe2024 augments the trunk input with geometric information (e.g., SDF) and employs a two-stage process: in Stage 1, both branch and trunk networks extract features using conventional ReLU activations; in Stage 2, the branch network continues with ReLU while the trunk network refines its features using sinusoidal (SIREN) activations, enabling more accurate capture of complex high frequency geometric details.
• Figure 4: Distribution of Reynolds numbers across different train-test splitting strategies. The left column represents the easy case, while the right column corresponds to the hard case. The top row shows the training dataset, and the bottom row shows the test dataset. The baseline random split ensures a uniform distribution of Reynolds numbers in both train and test sets, while the extrapolatory split reserves the highest 20% of Reynolds numbers for testing, with the training set containing only the lower 80%.
• Figure 5: Visualization of the loss functions in a 3×3×3 grid. $L1$ and $L2$ evaluate errors at element centers, while $L3$ and $L4$ enforce velocity consistency at boundary points while keeping gradient losses at element centers.