# Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries

arXiv: https://arxiv.org/abs/2502.02414 · HTML: https://mlsanity.com/paper/2502.02414/

## Key facts

- Cited by 97 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Feb 2025.

## Built on by (most cited first, 30 of 97)

- [AB-UPT: Scaling Neural CFD Surrogates for High-Fidelity Automotive Aerodynamics Simulations via Anchored-Branched Universal Physics Transformers](https://mlsanity.com/paper/2502.09692/) (2025)
- [Erwin: A Tree-based Hierarchical Transformer for Large-scale Physical Systems](https://mlsanity.com/paper/2502.17019/) (2025)
- [Transolver-3: Scaling Up Transformer Solvers to Industrial-Scale Geometries](https://mlsanity.com/paper/2602.04940/) (2026)
- [Transolver is a Linear Transformer: Revisiting Physics-Attention through the Lens of Linear Attention](https://mlsanity.com/paper/2511.06294/) (2025)
- [Automotive Crash Dynamics Modeling Accelerated with Machine Learning](https://mlsanity.com/paper/2510.15201/) (2025)
- [CarBench: A Comprehensive Benchmark for Neural Surrogates on High-Fidelity 3D Car Aerodynamics](https://mlsanity.com/paper/2512.07847/) (2025)
- [MeshGraphNet-Transformer: Scalable Mesh-based Learned Simulation for Solid Mechanics](https://mlsanity.com/paper/2601.23177/) (2026)
- [GeoPT: Scaling Physics Simulation via Lifted Geometric Pre-Training](https://mlsanity.com/paper/2602.20399/) (2026)
- [Structure-Preserving Learning Improves Geometry Generalization in Neural PDEs](https://mlsanity.com/paper/2602.02788/) (2026)
- [Enabling Automatic Differentiation with Mollified Graph Neural Operators](https://mlsanity.com/paper/2504.08277/) (2025)
- [Leveraging Operator Learning to Accelerate Convergence of the Preconditioned Conjugate Gradient Method](https://mlsanity.com/paper/2508.00101/) (2025)
- [FLARE: Fast Low-rank Attention Routing Engine](https://mlsanity.com/paper/2508.12594/) (2025)
- [EddyFormer: Accelerated Neural Simulations of Three-Dimensional Turbulence at Scale](https://mlsanity.com/paper/2510.24173/) (2025)
- [Going with the Speed of Sound: Pushing Neural Surrogates into Highly-turbulent Transonic Regimes](https://mlsanity.com/paper/2511.21474/) (2025)
- [A Neural-Operator Preconditioned Newton Method for Accelerated Nonlinear Solvers](https://mlsanity.com/paper/2511.08811/) (2025)
- [SMART: Scalable Mesh-free Aerodynamic Simulations from Raw Geometries using a Transformer-based Surrogate Model](https://mlsanity.com/paper/2601.18707/) (2026)
- [Simple yet Effective: Low-Rank Spatial Attention for Neural Operators](https://mlsanity.com/paper/2604.03582/) (2026)
- [Self-composing neural operators for high-frequency and multiscale PDE surrogates](https://mlsanity.com/paper/2508.20650/) (2025)
- [MSPT: Efficient Large-Scale Physical Modeling via Parallelized Multi-Scale Attention](https://mlsanity.com/paper/2512.01738/) (2025)
- [ArGEnT: Arbitrary Geometry-encoded Transformer for Operator Learning](https://mlsanity.com/paper/2602.11626/) (2026)
- [GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2508.04463/) (2025)
- [ReBaNO: Reduced Basis Neural Operator Mitigating Generalization Gaps and Achieving Discretization Invariance](https://mlsanity.com/paper/2509.09611/) (2025)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [Graph Deep Learning for Intracranial Aneurysm Blood Flow Simulation and Risk Assessment](https://mlsanity.com/paper/2512.09013/) (2025)
- [Flowers: A Warp Drive for Neural PDE Solvers](https://mlsanity.com/paper/2603.04430/) (2026)
- [Anchored-Branched Steady-state WInd Flow Transformer (AB-SWIFT): a metamodel for 3D atmospheric flow in urban environments](https://mlsanity.com/paper/2603.25635/) (2026)
- [Geometry-Aware Neural Optimizer for Shape Optimization and Inversion](https://mlsanity.com/paper/2605.04474/) (2026)
- [Neural-Schwarz Tiling for Geometry-Universal PDE Solving at Scale](https://mlsanity.com/paper/2605.12343/) (2026)
- [DrivAer Transformer: A high-precision and fast prediction method for vehicle aerodynamic drag coefficient based on the DrivAerNet++ dataset](https://mlsanity.com/paper/2504.08217/) (2025)
- [Adaptive Physics Transformer with Fused Global-Local Attention for Subsurface Energy Systems](https://mlsanity.com/paper/2602.11208/) (2026)

## Abstract (arXiv)

Although deep models have been widely explored in solving partial differential equations (PDEs), previous works are primarily limited to data only with up to tens of thousands of mesh points, far from the million-point scale required by industrial simulations that involve complex geometries. In the spirit of advancing neural PDE solvers to real industrial applications, we present Transolver++, a highly parallel and efficient neural solver that can accurately solve PDEs on million-scale geometries. Building upon previous advancements in solving PDEs by learning physical states via Transolver, Transolver++ is further equipped with an extremely optimized parallelism framework and a local adaptive mechanism to efficiently capture eidetic physical states from massive mesh points, successfully tackling the thorny challenges in computation and physics learning when scaling up input mesh size. Transolver++ increases the single-GPU input capacity to million-scale points for the first time and is capable of continuously scaling input size in linear complexity by increasing GPUs. Experimentally, Transolver++ yields 13% relative promotion across six standard PDE benchmarks and achieves over 20% performance gain in million-scale high-fidelity industrial simulations, whose sizes are 100× larger than previous benchmarks, covering car and 3D aircraft designs.