# Monotone Peridynamic Neural Operator for Nonlinear Material Modeling with Conditionally Unique Solutions

arXiv: https://arxiv.org/abs/2505.01060 · HTML: https://mlsanity.com/paper/2505.01060/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 22 of its 53 references that are in the map.
- First posted on arXiv 2 May 2025.

## Builds on (most cited first, 22 of 22)

- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [Deep Potential Molecular Dynamics: a scalable model with the accuracy of quantum mechanics](https://mlsanity.com/paper/1707.09571/) (2017)
- [Ab-Initio Solution of the Many-Electron Schr"odinger Equation with Deep Neural Networks](https://mlsanity.com/paper/1909.02487/) (2019)
- [Deep Neural Networks Motivated by Partial Differential Equations](https://mlsanity.com/paper/1804.04272/) (2018)
- [Integrating Neural Operators with Diffusion Models Improves Spectral Representation in Turbulence Modeling](https://mlsanity.com/paper/2409.08477/) (2024)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials](https://mlsanity.com/paper/1807.09829/) (2018)
- [Data-driven Tissue Mechanics with Polyconvex Neural Ordinary Differential Equations](https://mlsanity.com/paper/2110.03774/) (2021)
- [INO: Invariant Neural Operators for Learning Complex Physical Systems with Momentum Conservation](https://mlsanity.com/paper/2212.14365/) (2022)
- [On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling](https://mlsanity.com/paper/2109.11028/) (2021)
- [Harnessing the Power of Neural Operators with Automatically Encoded Conservation Laws](https://mlsanity.com/paper/2312.11176/) (2023)
- [Peridynamic Neural Operators: A Data-Driven Nonlocal Constitutive Model for Complex Material Responses](https://mlsanity.com/paper/2401.06070/) (2024)
- [Nonlocal Attention Operator: Materializing Hidden Knowledge Towards Interpretable Physics Discovery](https://mlsanity.com/paper/2408.07307/) (2024)
- [A data-driven peridynamic continuum model for upscaling molecular dynamics](https://mlsanity.com/paper/2108.04883/) (2021)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [Nonparametric learning of kernels in nonlocal operators](https://mlsanity.com/paper/2205.11006/) (2022)
- [Learning Memory and Material Dependent Constitutive Laws](https://mlsanity.com/paper/2502.05463/) (2025)
- [Heterogeneous Peridynamic Neural Operators: Discover Biotissue Constitutive Law and Microstructure From Digital Image Correlation Measurements](https://mlsanity.com/paper/2403.18597/) (2024)
- [Towards a unified nonlocal, peridynamics framework for the coarse-graining of molecular dynamics data with fractures](https://mlsanity.com/paper/2301.04540/) (2023)
- [An End-to-End Deep Learning Method for Solving Nonlocal Allen-Cahn and Cahn-Hilliard Phase-Field Models](https://mlsanity.com/paper/2410.08914/) (2024)

## Built on by (most cited first, 3 of 3)

- [Neural Interpretable PDEs: Harmonizing Fourier Insights with Attention for Scalable and Interpretable Physics Discovery](https://mlsanity.com/paper/2505.23106/) (2025)
- [A physics-guided smoothing method for material modeling with digital image correlation (DIC) measurements](https://mlsanity.com/paper/2505.18784/) (2025)
- [Well-posedness of neural turbulence closures and tangent dissipation](https://mlsanity.com/paper/2609.19647/) (2026)

## Abstract (arXiv)

Data-driven methods have emerged as powerful tools for modeling the responses of complex nonlinear materials directly from experimental measurements. Among these methods, the data-driven constitutive models present advantages in physical interpretability and generalizability across different boundary conditions/domain settings. However, the well-posedness of these learned models is generally not guaranteed a priori, which makes the models prone to non-physical solutions in downstream simulation tasks. In this study, we introduce monotone peridynamic neural operator (MPNO), a novel data-driven nonlocal constitutive model learning approach based on neural operators. Our approach learns a nonlocal kernel together with a nonlinear constitutive relation, while ensuring solution uniqueness through a monotone gradient network. This architectural constraint on gradient induces convexity of the learnt energy density function, thereby guaranteeing solution uniqueness of MPNO in small deformation regimes. To validate our approach, we evaluate MPNO's performance on both synthetic and real-world datasets. On synthetic datasets with manufactured kernel and constitutive relation, we show that the learnt model converges to the ground-truth as the measurement grid size decreases both theoretically and numerically. Additionally, our MPNO exhibits superior generalization capabilities than the conventional neural networks: it yields smaller displacement solution errors in down-stream tasks with new and unseen loadings. Finally, we showcase the practical utility of our approach through applications in learning a homogenized model from molecular dynamics data, highlighting its expressivity and robustness in real-world scenarios.