Geometry-Aware Neural Operators

Physically based two-dimensional hydraulic models are too slow for real-time inundation forecasting and large ensembles. We present a time-dependent geometry-informed neural operator (GINO) surrogate of LISFLOOD-FP for the municipality of Forlì (Italy), severely affected by the May 2023 Emilia-Romagna floods. The surrogate is designed for an operational setting in which only river stage gauges and rain gauges are available. A reproducible pipeline turns public regional observations into a physically based training set: each river inflow is driven by the first stage gauge downstream of it through a calibrated water-surface boundary condition, and rain enters as SCS-CN effective rainfall. The surrogate is an autoregressive GINO trained on 216 simulations of 24 observed events with scaled forcing. It predicts volume-conserving polygon-mean depths on an exposure-based Voronoi mesh of 659 nodes, with a 20 min step over 72 h. The surrogate is tested on 38 independent observed events that it never saw during training. On these events it reaches a critical success index (CSI) of 0.934 at 0.05m and 0.844 at 0.3 m, a depth RMSE of 0.008m and an $R^2$ of 0.984 for maximum-depth maps. On one NVIDIA A10G GPU a 72-h forecast takes 1.7 s instead of 55 s with LISFLOOD-FP, and a 50-member ensemble, computed in a single batched rollout, takes 6.2 s instead of about 46 min, which makes probabilistic flood forecasting practical. These results show that neural-operator surrogates can deliver accurate pluvial-fluvial flood forecasts driven by river stage for operational early warning.

Geometry-aware neural operators face substantial computational and memory demands when learning across complex physical domains. We introduce Architecture-Aware Mixed Precision (AA-MP), a replica-exchange training strategy that allocates numerical precision according to replica roles and architecture-specific operations. By retaining FP32 for refinement while selectively employing BF16 for exploration, AA-MP preserves predictive accuracy while improving training efficiency. Across fluid dynamics and three-dimensional structural mechanics benchmarks, AA-MP achieves comparable accuracy to full-precision training over an ensemble of runs, while reducing hot-replica peak GPU memory by 46.9% and training time by up to 25.8%. It also outperforms naive Automatic Mixed Precision (AMP) in memory efficiency and computational cost, with increasing memory advantages at larger model and geometry scales. Ablation studies show that retaining FP32 in the cold-replica is important for predictive accuracy, while additional FP32 operations in the exploratory replica provide limited benefits relative to their computational cost. Finally, conformal prediction for uncertainty calibration shows that AA-MP achieves near-nominal coverage with narrower prediction intervals than the alternatives. These results advance the scalability of geometry-aware scientific machine learning by reducing computational costs without compromising predictive accuracy or uncertainty quantification.

The geometry of latent representations governs which components should interact and how information should propagate, making geometry-aware operator design a fundamental ingredient of structured deep representation learning. However, existing neural architectures typically rely on generic operator templates or geometry-specific constructions, creating a need for a unified framework that can derive admissible operators directly from fixed structural information while remaining adaptive to changing contexts. We introduce Geometry-Induced Operator Families (GIOF), a general framework that converts fixed geometry into a structured family of propagation operators and dynamically selects an appropriate member of this family according to the current context. GIOF first transforms geometry-derived interaction channels into reusable generator bases, then combines them through a context-dependent selector and adaptive propagation scale, and finally realizes the selected operator through stable continuous-time propagation and a bottleneck residual layer. We establish theoretical guarantees covering parameter compression, identifiability, stability, locality, compositional structure, and oversmoothing behavior, while controlled experiments validate these mechanisms and experiments on PEMS-BAY and METR-LA achieve the lowest mean MAE across all reported regional-outage settings, improving over the strongest retained baseline by 2.4%--8.8% at 30% missing sensors.

While neural PDE solvers have demonstrated significant potential for accelerating engineering simulations, existing architectures remain constrained by high memory consumption and the single node bottleneck, where the maximum processable mesh resolution is strictly limited by the VRAM of a single compute unit. To address these challenges, we propose PGD-NO, a neural operator with Precomputed Geometry Decomposition, that relocates the computational overhead of geometric encoding to a deterministic pre-computation phase. By utilizing an iterative geometry decomposition algorithm to extract geometry tokens, our model decouples feature extraction from solution querying. This architecture enables linear memory scalability, allowing high fidelity learning on meshes exceeding 10 million nodes, a scale where existing architectures typically encounter memory exhaustion. PGD-NO demonstrates competitive predictive accuracy across diverse industrial benchmarks and provides intrinsic interpretability through attention mechanisms. By effectively overcoming traditional mesh-size constraints, PGD-NO offers a robust and efficient solution for the next generation of large-scale, high-fidelity industrial design applications.

Automotive crashworthiness optimization remains a safety-critical challenge, requiring the management of large-scale nonlinear structural deformations and energy dissipation through iterative, high-fidelity simulations. While traditional finite element solvers are computationally prohibitive, emerging operator learning frameworks provide rapid surrogate predictions; however, applying them to industrial-scale crash analysis, where complex geometry, contact nonlinearities, and rapidly evolving transient deformation coexist, remains an open challenge. In this paper, we demonstrate that the GeoTransolver framework provides a viable solution for accurate, high-fidelity crash dynamics prediction at industrial scale. Benchmarked on complex bumper beam and full-vehicle crash datasets, GeoTransolver captures multi-scale geometric context and accurately resolves plastic deformation patterns as well as acceleration profiles at critical occupant locations. Beyond the architecture itself, we propose and systematically evaluate a suite of temporal prediction recipes, including one-shot, time-conditional, and autoregressive rollout strategies, demonstrating that the one-shot approach achieves state-of-the-art accuracy with significantly reduced training overhead and inference latency. As a secondary contribution, we introduce a Fast Low-rank Attention Routing Engine (FLARE)-based modification to the GeoTransolver attention backbone that reduces memory overhead by approximately 2x while further improving predictive accuracy for O(N) long-range, high-frequency transients, preserving the geometry-aware cross-attention strengths of the base framework. Our results highlight the practical viability of geometry-aware operator learning for high-fidelity surrogate modeling of complex, safety-critical automotive dynamics.

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.

Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient plug-and-play module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50% at the cost of 2.26% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.

While Transformers have demonstrated remarkable potential in modeling Partial Differential Equations (PDEs), modeling large-scale unstructured meshes with complex geometries remains a significant challenge. Existing efficient architectures often employ feature dimensionality reduction strategies, which inadvertently induces Geometric Aliasing, resulting in the loss of critical physical boundary information. To address this, we propose the Physics-Geometry Operator Transformer (PGOT), designed to reconstruct physical feature learning through explicit geometry awareness. Specifically, we propose Spectrum-Preserving Geometric Attention (SpecGeo-Attention). Utilizing a ``physics slicing-geometry injection" mechanism, this module incorporates multi-scale geometric encodings to explicitly preserve multi-scale geometric features while maintaining linear computational complexity $O(N)$. Furthermore, PGOT dynamically routes computations to low-order linear paths for smooth regions and high-order non-linear paths for shock waves and discontinuities based on spatial coordinates, enabling spatially adaptive and high-precision physical field modeling. PGOT achieves consistent state-of-the-art performance across four standard benchmarks and excels in large-scale industrial tasks including airfoil and car designs.