# A Physics-Informed Fourier-Wavelet Transformer for Multiscale Computational Fluid Dynamics Surrogate Modeling

arXiv: https://arxiv.org/abs/2606.24696 · HTML: https://mlsanity.com/paper/2606.24696/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 32 of its 49 references that are in the map.
- First posted on arXiv 23 Jun 2026.

## Builds on (most cited first, 30 of 32)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Transolver: A Fast Transformer Solver for PDEs on General Geometries](https://mlsanity.com/paper/2402.02366/) (2024)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [BERT: A Review of Applications in Natural Language Processing and Understanding](https://mlsanity.com/paper/2103.11943/) (2021)
- [Physics-Informed Machine Learning: A Survey on Problems, Methods and Applications](https://mlsanity.com/paper/2211.08064/) (2022)
- [Operator Learning with Neural Fields: Tackling PDEs on General Geometries](https://mlsanity.com/paper/2306.07266/) (2023)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [Diffusion-Based Adversarial Sample Generation for Improved Stealthiness and Controllability](https://mlsanity.com/paper/2305.16494/) (2023)
- [PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networks](https://mlsanity.com/paper/2307.11833/) (2023)
- [Eagle: Large-Scale Learning of Turbulent Fluid Dynamics with Mesh Transformers](https://mlsanity.com/paper/2302.10803/) (2023)
- [A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions](https://mlsanity.com/paper/2302.14227/) (2023)
- [CFDBench: A Large-Scale Benchmark for Machine Learning Methods in Fluid Dynamics](https://mlsanity.com/paper/2310.05963/) (2023)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Hierarchical Deep Learning of Multiscale Differential Equation Time-Steppers](https://mlsanity.com/paper/2008.09768/) (2020)
- [Wavelet Diffusion Neural Operator](https://mlsanity.com/paper/2412.04833/) (2024)
- [Recent Advances on Machine Learning for Computational Fluid Dynamics: A Survey](https://mlsanity.com/paper/2408.12171/) (2024)
- [Toward a Better Understanding of Fourier Neural Operators from a Spectral Perspective](https://mlsanity.com/paper/2404.07200/) (2024)
- [Physics Informed Token Transformer for Solving Partial Differential Equations](https://mlsanity.com/paper/2305.08757/) (2023)
- [Physics-Informed Geometry-Aware Neural Operator](https://mlsanity.com/paper/2408.01600/) (2024)
- [RealPDEBench: A Benchmark for Complex Physical Systems with Real-World Data](https://mlsanity.com/paper/2601.01829/) (2026)
- [Multi-scale Time-stepping of Partial Differential Equations with Transformers](https://mlsanity.com/paper/2311.02225/) (2023)
- [Masked Autoencoders are PDE Learners](https://mlsanity.com/paper/2403.17728/) (2024)
- [Challenges and Advancements in Modeling Shock Fronts with Physics-Informed Neural Networks: A Review and Benchmarking Study](https://mlsanity.com/paper/2503.17379/) (2025)
- [MeshMask: Physics-Based Simulations with Masked Graph Neural Networks](https://mlsanity.com/paper/2501.08738/) (2025)
- [Physics-Informed Neural Networks and Extensions](https://mlsanity.com/paper/2408.16806/) (2024)
- [FourierFlow: Frequency-aware Flow Matching for Generative Turbulence Modeling](https://mlsanity.com/paper/2506.00862/) (2025)
- [On the Effectiveness of Neural Operators at Zero-Shot Weather Downscaling](https://mlsanity.com/paper/2409.13955/) (2024)

## Built on by (most cited first, 2 of 2)

- [Modular Deep Learning Mechanisms for Auditable Next-Day Wildfire Spread Prediction](https://mlsanity.com/paper/2609.17763/) (2026)
- [Do Better Scores Mean Better Physics? Physics-Grounded Explanations for Sim2Real Neural Operators](https://mlsanity.com/paper/2610.00415/) (2026)

## Abstract (arXiv)

Physics-informed surrogate models can accelerate computational fluid dynamics simulations. However, many existing methods reproduce global flow patterns more reliably than localized multiscale structures. This study presents a physics-informed Fourier-wavelet transformer for next-step velocity-field reconstruction in real-world flow benchmarks. The proposed formulation combines hybrid Fourier-wavelet spectral encoding with physics-biased self-attention based on partial differential equation residual diagnostics. It also uses self-supervised pretraining through Masked Physics Prediction and Equation Consistency Prediction. The experiments are conducted on two real benchmark cases: cylinder-wake flow and fluid-structure interaction. All approaches are evaluated under a shared local protocol and compared with spectral, transformer-based, operator-learning, and physics-informed neural-network baselines. On the cylinder-wake benchmark, the proposed model achieves the best aggregate accuracy, with an all-channel normalized mean-squared error of 0.05875 and an all-channel Pearson correlation coefficient of 0.97019. On the fluid-structure-interaction benchmark, it gives the lowest all-channel normalized mean-squared error of 2.70 × 10-4, compared with 4.02 × 10-4 for the strongest baseline. Component-wise field comparisons and scale-separated diagnostics further show stronger recovery of localized wake structures, including near-body, wake-core, and far-wake features. The results demonstrate improved real-world flow reconstruction while maintaining a practical accuracy-cost tradeoff.