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

2025-05-30

AI Summary

Accurate and interpretable modeling of the nonlinear voltage–displacement hysteresis in piezoelectric systems remains challenging; existing neural operator methods suffer from opacity and poor generalization across voltage domains. Method: We propose the first neuro-symbolic operator framework, integrating Fourier neural operators (FNOs) with sparse identification of nonlinear dynamics (SINDy)-driven symbolic regression to jointly learn robust frequency-domain features and automatically discover analytic hysteresis equations. Contribution/Results: The resulting white-box model achieves high accuracy, strong interpretability, and cross-distribution generalization. It faithfully reconstructs complex hysteresis loops—including butterfly-shaped curves—and maintains stable predictive performance under unseen voltage ranges, input noise, and low-fidelity inputs. Quantitatively, it significantly outperforms both pure neural operators and conventional symbolic regression methods, establishing a new paradigm for physics-informed, interpretable surrogate modeling of hysteretic systems.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Search and Optimization: Learning to Search

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

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.

Problem

• Addresses interpretability issues in neural operators for piezoelectric systems
• Enhances generalizability of neural operators to novel voltage conditions
• Provides robust hysteresis modeling for noisy and low-fidelity data

Innovation

• Neuro-symbolic operator for interpretable hysteresis modeling
• Fourier neural operator maps voltage to displacement
• Library-based sparse model discovery for white-box models