# Neuro-Symbolic Operator for Interpretable and Generalizable Characterization of Complex Piezoelectric Systems

arXiv: https://arxiv.org/abs/2505.24578 · HTML: https://mlsanity.com/paper/2505.24578/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 39 references that are in the map.
- First posted on arXiv 30 May 2025.

## Builds on (most cited first, 16 of 16)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (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)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Fourier Neural Operators for Arbitrary Resolution Climate Data Downscaling](https://mlsanity.com/paper/2305.14452/) (2023)
- [Sound propagation in realistic interactive 3D scenes with parameterized sources using deep neural operators](https://mlsanity.com/paper/2308.05141/) (2023)
- [Neural Operator: Is data all you need to model the world? An insight into the paradigm of data-driven scientific ML](https://mlsanity.com/paper/2301.13331/) (2023)
- [Magnetic Hysteresis Modeling with Neural Operators](https://mlsanity.com/paper/2407.03261/) (2024)
- [Discovery of sparse hysteresis models for piezoelectric materials](https://mlsanity.com/paper/2302.05313/) (2023)
- [Dynamic hysteresis model of grain-oriented ferromagnetic material using neural operators](https://mlsanity.com/paper/2504.04863/) (2025)
- [Beyond Accuracy: EcoL2 Metric for Sustainable Neural PDE Solvers](https://mlsanity.com/paper/2505.12556/) (2025)

## Abstract (arXiv)

Complex piezoelectric systems are foundational in industrial applications. Their performance, however, is challenged by the nonlinear voltage-displacement hysteretic relationships. Efficient characterization methods are, therefore, essential for reliable design, monitoring, and maintenance. Recently proposed neural operator methods serve as surrogates for system characterization but face two pressing issues: interpretability and generalizability. State-of-the-art (SOTA) neural operators are black-boxes, providing little insight into the learned operator. Additionally, generalizing them to novel voltages and predicting displacement profiles beyond the training domain is challenging, limiting their practical use. To address these limitations, this paper proposes a neuro-symbolic operator (NSO) framework that derives the analytical operators governing hysteretic relationships. NSO first learns a Fourier neural operator mapping voltage fields to displacement profiles, followed by a library-based sparse model discovery method, generating white-box parsimonious models governing the underlying hysteresis. These models enable accurate and interpretable prediction of displacement profiles across varying and out-of-distribution voltage fields, facilitating generalizability. The potential of NSO is demonstrated by accurately predicting voltage-displacement hysteresis, including butterfly-shaped relationships. Moreover, NSO predicts displacement profiles even for noisy and low-fidelity voltage data, emphasizing its robustness. The results highlight the advantages of NSO compared to SOTA neural operators and model discovery methods on several evaluation metrics. Consequently, NSO contributes to characterizing complex piezoelectric systems while improving the interpretability and generalizability of neural operators, essential for design, monitoring, maintenance, and other real-world scenarios.