# Latent Neural Operator for Solving Forward and Inverse PDE Problems

arXiv: https://arxiv.org/abs/2406.03923 · HTML: https://mlsanity.com/paper/2406.03923/

## Key facts

- Cited by 80 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Jun 2024.

## Built on by (most cited first, 30 of 80)

- [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)
- [Spatially-Aware Diffusion Models with Cross-Attention for Global Field Reconstruction with Sparse Observations](https://mlsanity.com/paper/2409.00230/) (2024)
- [Transolver is a Linear Transformer: Revisiting Physics-Attention through the Lens of Linear Attention](https://mlsanity.com/paper/2511.06294/) (2025)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space](https://mlsanity.com/paper/2410.11382/) (2024)
- [NeuralDEM -- Real-time Simulation of Industrial Particulate Flows](https://mlsanity.com/paper/2411.09678/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learning](https://mlsanity.com/paper/2509.12344/) (2025)
- [FLARE: Fast Low-rank Attention Routing Engine](https://mlsanity.com/paper/2508.12594/) (2025)
- [Foundation Models for Scientific Discovery: From Paradigm Enhancement to Paradigm Transition](https://mlsanity.com/paper/2510.15280/) (2025)
- [Neural Green's Functions](https://mlsanity.com/paper/2511.01924/) (2025)
- [Learning PDE Solvers with Physics and Data: A Unifying View of Physics-Informed Neural Networks and Neural Operators](https://mlsanity.com/paper/2601.14517/) (2026)
- [Latent Neural Operator Pretraining for Solving Time-Dependent PDEs](https://mlsanity.com/paper/2410.20100/) (2024)
- [DiffusionRollout: Uncertainty-Aware Rollout Planning in Long-Horizon PDE Solving](https://mlsanity.com/paper/2602.13616/) (2026)
- [FNOPE: Simulation-based inference on function spaces with Fourier Neural Operators](https://mlsanity.com/paper/2505.22573/) (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)
- [A Deep Learning approach for parametrized and time dependent Partial Differential Equations using Dimensionality Reduction and Neural ODEs](https://mlsanity.com/paper/2502.08683/) (2025)
- [MaNGO - Adaptable Graph Network Simulators via Meta-Learning](https://mlsanity.com/paper/2510.05874/) (2025)
- [MSPT: Efficient Large-Scale Physical Modeling via Parallelized Multi-Scale Attention](https://mlsanity.com/paper/2512.01738/) (2025)
- [Discovering Physical Directions in Weight Space: Composing Neural PDE Experts](https://mlsanity.com/paper/2605.14546/) (2026)
- [GFocal: A Global-Focal Neural Operator for Solving PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2508.04463/) (2025)
- [A unified physics-informed generative operator framework for general inverse problems](https://mlsanity.com/paper/2511.03241/) (2025)
- [Generalizing PDE Emulation with Equation-Aware Neural Operators](https://mlsanity.com/paper/2511.09729/) (2025)
- [Efficient Generative Transformer Operators For Million-Point PDEs](https://mlsanity.com/paper/2512.04974/) (2025)
- [From Basis to Basis: Gaussian Particle Representation for Interpretable PDE Operators](https://mlsanity.com/paper/2602.21551/) (2026)
- [Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations](https://mlsanity.com/paper/2605.13790/) (2026)
- [PDEInvBench: A Comprehensive Dataset and Design Space Exploration of Neural Networks for PDE Inverse Problems](https://mlsanity.com/paper/2605.25353/) (2026)
- [Autoregression-Free Neural Operators for Time-Dependent PDEs](https://mlsanity.com/paper/2605.25413/) (2026)

## Abstract (arXiv)

Neural operators effectively solve PDE problems from data without knowing the explicit equations, which learn the map from the input sequences of observed samples to the predicted values. Most existing works build the model in the original geometric space, leading to high computational costs when the number of sample points is large. We present the Latent Neural Operator (LNO) solving PDEs in the latent space. In particular, we first propose Physics-Cross-Attention (PhCA) transforming representation from the geometric space to the latent space, then learn the operator in the latent space, and finally recover the real-world geometric space via the inverse PhCA map. Our model retains flexibility that can decode values in any position not limited to locations defined in the training set, and therefore can naturally perform interpolation and extrapolation tasks particularly useful for inverse problems. Moreover, the proposed LNO improves both prediction accuracy and computational efficiency. Experiments show that LNO reduces the GPU memory by 50%, speeds up training 1.8 times, and reaches state-of-the-art accuracy on four out of six benchmarks for forward problems and a benchmark for inverse problem. Code is available at https://github.com/L-I-M-I-T/LatentNeuralOperator.