# Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning

arXiv: https://arxiv.org/abs/2407.15887 · HTML: https://mlsanity.com/paper/2407.15887/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 42 references that are in the map.
- First posted on arXiv 21 Jul 2024.

## Builds on (most cited first, 21 of 21)

- [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)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [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)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [A physics-informed variational DeepONet for predicting the crack path in brittle materials](https://mlsanity.com/paper/2108.06905/) (2021)
- [Improved architectures and training algorithms for deep operator networks](https://mlsanity.com/paper/2110.01654/) (2021)
- [Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling](https://mlsanity.com/paper/2207.02338/) (2022)
- [Separable Physics-Informed Neural Networks](https://mlsanity.com/paper/2306.15969/) (2023)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [On the influence of over-parameterization in manifold based surrogates and deep neural operators](https://mlsanity.com/paper/2203.05071/) (2022)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [Neural operator learning of heterogeneous mechanobiological insults contributing to aortic aneurysms](https://mlsanity.com/paper/2205.03780/) (2022)
- [Energy-Dissipative Evolutionary Deep Operator Neural Networks](https://mlsanity.com/paper/2306.06281/) (2023)
- [Advanced Deep Operator Networks to Predict Multiphysics Solution Fields in Materials Processing and Additive Manufacturing](https://mlsanity.com/paper/2403.14795/) (2024)
- [PGD reduced-order modeling for structural dynamics applications](https://mlsanity.com/paper/2301.10633/) (2023)

## Built on by (most cited first, 15 of 15)

- [Neural Chaos: A Spectral Stochastic Neural Operator](https://mlsanity.com/paper/2502.11835/) (2025)
- [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)
- [SPIKANs: Separable Physics-Informed Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2411.06286/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [Efficient Training of Deep Neural Operator Networks via Randomized Sampling](https://mlsanity.com/paper/2409.13280/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [A Physics-Informed Meta-Learning Framework for the Continuous Solution of Parametric PDEs on Arbitrary Geometries](https://mlsanity.com/paper/2504.02459/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [Hybrid coupling with operator inference and the overlapping Schwarz alternating method](https://mlsanity.com/paper/2511.20687/) (2025)
- [Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networks](https://mlsanity.com/paper/2503.17402/) (2025)
- [Physics-Informed DeepONet Coupled with FEM for Convective Transport in Porous Media with Sharp Gaussian Sources](https://mlsanity.com/paper/2508.19847/) (2025)
- [A Unified Generative-Predictive Framework for Deterministic Inverse Design](https://mlsanity.com/paper/2512.15746/) (2025)
- [An Attention-based Spatio-Temporal Neural Operator for Evolving Physics](https://mlsanity.com/paper/2506.11328/) (2025)
- [Label-Free Finite-Volume-Residual Training of Attention Graph Neural Networks for Coupled Thermo-Fluid Fields](https://mlsanity.com/paper/2607.20321/) (2026)

## Abstract (arXiv)

The deep operator network (DeepONet) is a popular neural operator architecture that has shown promise in solving partial differential equations (PDEs) by using deep neural networks to map between infinite-dimensional function spaces. In the absence of labeled datasets, we utilize the PDE residual loss to learn the physical system, an approach known as physics-informed DeepONet. This method faces significant computational challenges, primarily due to the curse of dimensionality, as the computational cost increases exponentially with finer discretization. In this paper, we introduce the Separable DeepONet framework to address these challenges and improve scalability for high-dimensional PDEs. Our approach involves a factorization technique where sub-networks handle individual one-dimensional coordinates, thereby reducing the number of forward passes and the size of the Jacobian matrix. By using forward-mode automatic differentiation, we further optimize the computational cost related to the Jacobian matrix. As a result, our modifications lead to a linear scaling of computational cost with discretization density, making Separable DeepONet suitable for high-dimensional PDEs. We validate the effectiveness of the separable architecture through three benchmark PDE models: the viscous Burgers equation, Biot's consolidation theory, and a parametrized heat equation. In all cases, our proposed framework achieves comparable or improved accuracy while significantly reducing computational time compared to conventional DeepONet. These results demonstrate the potential of Separable DeepONet in efficiently solving complex, high-dimensional PDEs, advancing the field of physics-informed machine learning.