# Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space

arXiv: https://arxiv.org/abs/2410.11382 · HTML: https://mlsanity.com/paper/2410.11382/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 36 of its 49 references that are in the map.
- First posted on arXiv 15 Oct 2024.

## Builds on (most cited first, 30 of 36)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Empirical Evaluation of Gated Recurrent Neural Networks on Sequence Modeling](https://mlsanity.com/paper/1412.3555/) (2014)
- [Transformers are RNNs: Fast Autoregressive Transformers with Linear Attention](https://mlsanity.com/paper/2006.16236/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [FNet: Mixing Tokens with Fourier Transforms](https://mlsanity.com/paper/2105.03824/) (2021)
- [Choose a Transformer: Fourier or Galerkin](https://mlsanity.com/paper/2105.14995/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [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)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [Spherical Fourier Neural Operators: Learning Stable Dynamics on the Sphere](https://mlsanity.com/paper/2306.03838/) (2023)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [DPOT: Auto-Regressive Denoising Operator Transformer for Large-Scale PDE Pre-Training](https://mlsanity.com/paper/2403.03542/) (2024)
- [Scalable Transformer for PDE Surrogate Modeling](https://mlsanity.com/paper/2305.17560/) (2023)
- [Multiple Physics Pretraining for Physical Surrogate Models](https://mlsanity.com/paper/2310.02994/) (2023)
- [Solving High-Dimensional PDEs with Latent Spectral Models](https://mlsanity.com/paper/2301.12664/) (2023)
- [Latent Neural Operator for Solving Forward and Inverse PDE Problems](https://mlsanity.com/paper/2406.03923/) (2024)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Bird's-Eye-View Scene Graph for Vision-Language Navigation](https://mlsanity.com/paper/2308.04758/) (2023)
- [Volumetric Environment Representation for Vision-Language Navigation](https://mlsanity.com/paper/2403.14158/) (2024)
- [Improved Operator Learning by Orthogonal Attention](https://mlsanity.com/paper/2310.12487/) (2023)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)

## Built on by (most cited first, 10 of 10)

- [LD-EnSF: Synergizing Latent Dynamics with Ensemble Score Filters for Fast Data Assimilation with Sparse Observations](https://mlsanity.com/paper/2411.19305/) (2024)
- [Simple yet Effective: Low-Rank Spatial Attention for Neural Operators](https://mlsanity.com/paper/2604.03582/) (2026)
- [3DID: Direct 3D Inverse Design for Aerodynamics with Physics-Aware Optimization](https://mlsanity.com/paper/2512.08987/) (2025)
- [Efficient High-Accuracy PDEs Solver with the Linear Attention Neural Operator](https://mlsanity.com/paper/2510.16816/) (2025)
- [Don't Fix the Basis -- Learn It: Spectral Representation with Adaptive Basis Learning for PDEs](https://mlsanity.com/paper/2605.10451/) (2026)
- [Harnessing AI for Inverse Partial Differential Equation Problems: Past, Present, and Prospects](https://mlsanity.com/paper/2605.16966/) (2026)
- [From Points to Edges: Edge-Conditioned Spectral Operators for Physics-Sensitive PDE Learning](https://mlsanity.com/paper/2608.06894/) (2026)
- [Kolmogorov-Arnold Representation for Symplectic Learning: Advancing Hamiltonian Neural Networks](https://mlsanity.com/paper/2508.19410/) (2025)
- [Transolver-σ: Joint Spectral-Physical Subspace Modeling for Neural PDE Solving](https://mlsanity.com/paper/2609.37279/) (2026)
- [Cluster Attention Neural Operators for Solving Parametric Partial Differential Equations](https://mlsanity.com/paper/2609.39914/) (2026)

## Abstract (arXiv)

Recent advances in operator learning have produced two distinct approaches for solving partial differential equations (PDEs): attention-based methods offering point-level adaptability but lacking spectral constraints, and spectral-based methods providing domain-level continuity priors but limited in local flexibility. This dichotomy has hindered the development of PDE solvers with both strong flexibility and generalization capability. This work introduces Holistic Physics Mixer (HPM), a simple framework that bridges this gap by integrating spectral and physical information in a unified space. HPM unifies both approaches as special cases while enabling more powerful spectral-physical interactions beyond either method alone. This enables HPM to inherit both the strong generalization of spectral methods and the flexibility of attention mechanisms while avoiding their respective limitations. Through extensive experiments across diverse PDE problems, we demonstrate that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency, while maintaining strong generalization capabilities with limited training data and excellent zero-shot performance on unseen resolutions.