# Physics-Guided Transformer (PGT): Physics-Aware Attention Mechanism for PINNs

arXiv: https://arxiv.org/abs/2603.27929 · HTML: https://mlsanity.com/paper/2603.27929/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 30 references that are in the map.
- First posted on arXiv 30 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [An Image is Worth 16x16 Words: Transformers for Image Recognition at Scale](https://mlsanity.com/paper/2010.11929/) (2020)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Multi-Task Learning Using Uncertainty to Weigh Losses for Scene Geometry and Semantics](https://mlsanity.com/paper/1705.07115/) (2017)
- [Group Equivariant Convolutional Networks](https://mlsanity.com/paper/1602.07576/) (2016)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Machine Learning for Fluid Mechanics](https://mlsanity.com/paper/1905.11075/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [Generalizing Convolutional Neural Networks for Equivariance to Lie Groups on Arbitrary Continuous Data](https://mlsanity.com/paper/2002.12880/) (2020)
- [Understanding and mitigating gradient pathologies in physics-informed neural networks](https://mlsanity.com/paper/2001.04536/) (2020)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)

## Abstract (arXiv)

Reconstructing continuous physical fields from sparse, irregular observations is a central challenge in scientific machine learning, particularly for systems governed by partial differential equations (PDEs). Existing physics-informed methods typically enforce governing equations as soft penalty terms during optimization, often leading to gradient imbalance, instability, and degraded physical consistency under limited data. We introduce the Physics-Guided Transformer (PGT), a neural architecture that embeds physical structure directly into the self-attention mechanism. Specifically, PGT incorporates a heat-kernel-derived additive bias into attention logits, encoding diffusion dynamics and temporal causality within the representation. Query coordinates attend to these physics-conditioned context tokens, and the resulting features are decoded using a FiLM-modulated sinusoidal implicit network that adaptively controls spectral response. We evaluate PGT on the one-dimensional heat equation and two-dimensional incompressible Navier-Stokes systems. In sparse 1D reconstruction with 100 observations, PGT achieves a relative L2 error of 5.9e-3, significantly outperforming both PINNs and sinusoidal representations. In the 2D cylinder wake problem, PGT uniquely achieves both low PDE residual (8.3e-4) and competitive relative error (0.034), outperforming methods that optimize only one objective. These results demonstrate that embedding physics within attention improves stability, generalization, and physical fidelity under data-scarce conditions.