# PMNO: A novel physics guided multi-step neural operator predictor for partial differential equations

arXiv: https://arxiv.org/abs/2506.01598 · HTML: https://mlsanity.com/paper/2506.01598/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 67 references that are in the map.
- First posted on arXiv 2 Jun 2025.

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

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [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)
- [Convolutional LSTM Network: A Machine Learning Approach for Precipitation Nowcasting](https://mlsanity.com/paper/1506.04214/) (2015)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [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)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Physics-informed neural networks with hard constraints for inverse design](https://mlsanity.com/paper/2102.04626/) (2021)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [FractalNet: Ultra-Deep Neural Networks without Residuals](https://mlsanity.com/paper/1605.07648/) (2016)
- [Beyond Finite Layer Neural Networks: Bridging Deep Architectures and Numerical Differential Equations](https://mlsanity.com/paper/1710.10121/) (2017)
- [Theory-guided Data Science: A New Paradigm for Scientific Discovery from Data](https://mlsanity.com/paper/1612.08544/) (2016)
- [Data-Driven Forecasting of High-Dimensional Chaotic Systems with Long Short-Term Memory Networks](https://mlsanity.com/paper/1802.07486/) (2018)
- [Machine Learning of Linear Differential Equations using Gaussian Processes](https://mlsanity.com/paper/1701.02440/) (2017)
- [Transformers for Modeling Physical Systems](https://mlsanity.com/paper/2010.03957/) (2020)
- [A Method for Representing Periodic Functions and Enforcing Exactly Periodic Boundary Conditions with Deep Neural Networks](https://mlsanity.com/paper/2007.07442/) (2020)
- [Wavelet neural operator: a neural operator for parametric partial differential equations](https://mlsanity.com/paper/2205.02191/) (2022)
- [Encoding physics to learn reaction-diffusion processes](https://mlsanity.com/paper/2106.04781/) (2021)
- [Multi-level Residual Networks from Dynamical Systems View](https://mlsanity.com/paper/1710.10348/) (2017)
- [Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equations](https://mlsanity.com/paper/2203.01360/) (2022)
- [Evolutional Deep Neural Network](https://mlsanity.com/paper/2103.09959/) (2021)
- [PolyNet: A Pursuit of Structural Diversity in Very Deep Networks](https://mlsanity.com/paper/1611.05725/) (2016)
- [Neural Operator Learning for Long-Time Integration in Dynamical Systems with Recurrent Neural Networks](https://mlsanity.com/paper/2303.02243/) (2023)
- [Stochastic Taylor Derivative Estimator: Efficient amortization for arbitrary differential operators](https://mlsanity.com/paper/2412.00088/) (2024)
- [A MgNO Method for Multiphase Flow in Porous Media](https://mlsanity.com/paper/2407.02505/) (2024)

## Built on by (most cited first, 1 of 1)

- [SpectONet: A Physics-Guided Spectral Deep Operator Network for Euler-Bernoulli Beam Dynamics](https://mlsanity.com/paper/2607.25790/) (2026)

## Abstract (arXiv)

Neural operators, which aim to approximate mappings between infinite-dimensional function spaces, have been widely applied in the simulation and prediction of physical systems. However, the limited representational capacity of network architectures, combined with their heavy reliance on large-scale data, often hinder effective training and result in poor extrapolation performance. In this paper, inspired by traditional numerical methods, we propose a novel physics guided multi-step neural operator (PMNO) architecture to address these challenges in long-horizon prediction of complex physical systems. Distinct from general operator learning methods, the PMNO framework replaces the single-step input with multi-step historical data in the forward pass and introduces an implicit time-stepping scheme based on the Backward Differentiation Formula (BDF) during backpropagation. This design not only strengthens the model's extrapolation capacity but also facilitates more efficient and stable training with fewer data samples, especially for long-term predictions. Meanwhile, a causal training strategy is employed to circumvent the need for multi-stage training and to ensure efficient end-to-end optimization. The neural operator architecture possesses resolution-invariant properties, enabling the trained model to perform fast extrapolation on arbitrary spatial resolutions. We demonstrate the superior predictive performance of PMNO predictor across a diverse range of physical systems, including 2D linear system, modeling over irregular domain, complex-valued wave dynamics, and reaction-diffusion processes. Depending on the specific problem setting, various neural operator architectures, including FNO, DeepONet, and their variants, can be seamlessly integrated into the PMNO framework.