# Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs

arXiv: https://arxiv.org/abs/2501.08428 · HTML: https://mlsanity.com/paper/2501.08428/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 40 of its 69 references that are in the map.
- First posted on arXiv 14 Jan 2025.

## Builds on (most cited first, 30 of 40)

- [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)
- [A Review of Uncertainty Quantification in Deep Learning: Techniques, Applications and Challenges](https://mlsanity.com/paper/2011.06225/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [FourCastNet: Accelerating Global High-Resolution Weather Forecasting using Adaptive Fourier Neural Operators](https://mlsanity.com/paper/2208.05419/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [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)
- [Learning two-phase microstructure evolution using neural operators and autoencoder architectures](https://mlsanity.com/paper/2204.07230/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Novel DeepONet architecture to predict stresses in elastoplastic structures with variable complex geometries and loads](https://mlsanity.com/paper/2306.03645/) (2023)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Physics informed WNO](https://mlsanity.com/paper/2302.05925/) (2023)
- [Semi-supervised Invertible Neural Operators for Bayesian Inverse Problems](https://mlsanity.com/paper/2209.02772/) (2022)
- [Invertible Fourier Neural Operators for Tackling Both Forward and Inverse Problems](https://mlsanity.com/paper/2402.11722/) (2024)
- [Synergistic Learning with Multi-Task DeepONet for Efficient PDE Problem Solving](https://mlsanity.com/paper/2408.02198/) (2024)
- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Learning High-Dimensional Parametric Maps via Reduced Basis Adaptive Residual Networks](https://mlsanity.com/paper/2112.07096/) (2021)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [MultiAuto-DeepONet: A Multi-resolution Autoencoder DeepONet for Nonlinear Dimension Reduction, Uncertainty Quantification and Operator Learning of Forward and Inverse Stochastic Problems](https://mlsanity.com/paper/2204.03193/) (2022)
- [Physics-informed active learning with simultaneous weak-form latent space dynamics identification](https://mlsanity.com/paper/2407.00337/) (2024)
- [Physics-Informed Deep Inverse Operator Networks for Solving PDE Inverse Problems](https://mlsanity.com/paper/2412.03161/) (2024)
- [Developing a cost-effective emulator for groundwater flow modeling using deep neural operators](https://mlsanity.com/paper/2304.12299/) (2023)

## Built on by (most cited first, 9 of 9)

- [Geometry-Aware Neural Optimizer for Shape Optimization and Inversion](https://mlsanity.com/paper/2605.04474/) (2026)
- [Di-BiLPS: Denoising induced Bidirectional Latent-PDE-Solver under Sparse Observations](https://mlsanity.com/paper/2605.13790/) (2026)
- [Reduced Order Modeling of Energetic Materials Using Physics-Aware Recurrent Convolutional Neural Networks in a Latent Space (LatentPARC)](https://mlsanity.com/paper/2509.12401/) (2025)
- [Pseudo-differential-enhanced physics-informed neural networks](https://mlsanity.com/paper/2602.14663/) (2026)
- [Latent PDE mapping for efficient physics-informed learning across geometries with limited data](https://mlsanity.com/paper/2607.22215/) (2026)
- [SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamics](https://mlsanity.com/paper/2607.25790/) (2026)
- [Geometry-aware Incremental Neural Operator for Long-Horizon PDE prediction](https://mlsanity.com/paper/2608.11237/) (2026)
- [HiLNO: A Hierarchical Latent Neural Operator with Multi-Scale Supervision for PDEs on General Geometries](https://mlsanity.com/paper/2609.18419/) (2026)
- [Hypernetwork-Parameterized Spatially Adaptive Neural Operators for PDE Learning](https://mlsanity.com/paper/2609.20309/) (2026)

## Abstract (arXiv)

Deep operator network (DeepONet) has shown significant promise as surrogate models for systems governed by partial differential equations (PDEs), enabling accurate mappings between infinite-dimensional function spaces. However, when applied to systems with high-dimensional input-output mappings arising from large numbers of spatial and temporal collocation points, these models often require heavily overparameterized networks, leading to long training times. Latent DeepONet addresses some of these challenges by introducing a two-step approach: first learning a reduced latent space using a separate model, followed by operator learning within this latent space. While efficient, this method is inherently data-driven and lacks mechanisms for incorporating physical laws, limiting its robustness and generalizability in data-scarce settings. In this work, we propose PI-Latent-NO, a physics-informed latent neural operator framework that integrates governing physics directly into the learning process. Our architecture features two coupled DeepONets trained end-to-end: a Latent-DeepONet that learns a low-dimensional representation of the solution, and a Reconstruction-DeepONet that maps this latent representation back to the physical space. By embedding PDE constraints into the training via automatic differentiation, our method eliminates the need for labeled training data and ensures physics-consistent predictions. The proposed framework is both memory and compute-efficient, exhibiting near-constant scaling with problem size and demonstrating significant speedups over traditional physics-informed operator models. We validate our approach on a range of parametric PDEs, showcasing its accuracy, scalability, and suitability for real-time prediction in complex physical systems.