Fuzzy Neural Network Performance and Interpretability of Quantum Wavefunction Probability Predictions

2025/11/07 by Pedro H. M. Zanineli, Zanineli, Pedro H. M., Monteiro, Matheus Zaia +6
Materials Science · Physics and Astronomy · #Chemical Physics (physics.chem-ph) #Data Analysis #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Machine Learning in Materials Science #Materials Science (cond-mat.mtrl-sci) #Quantum many-body systems #Quantum, superfluid, helium dynamics #Statistics and Probability (physics.data-an)

paper · pdf · doi:10.48550/arxiv.2511.05261

openalex publication_date 2025/11/07 · openalex created_date 2025/11/11 · openalex updated_date 2026/07/28

Abstract

Predicting quantum wavefunction probability distributions is crucial for computational chemistry and materials science, yet machine learning (ML) models often face a trade-off between accuracy and interpretability. This study compares Artificial Neural Networks (ANNs) and Adaptive Neuro-Fuzzy Inference Systems (ANFIS) in modeling quantum probability distributions for the H₂⁺ ion, leveraging data generated via Physics-Informed Neural Networks (PINNs). While ANN achieved superior accuracy (R² = 0.99 vs ANFIS's 0.95 with Gaussian membership functions), it required over 50x more parameters (2,305 vs 39-45). ANFIS, however, provided unique interpretability: its Gaussian membership functions encoded spatial electron localization near proton positions (μ= 1.2 A), mirroring Born probability densities, while fuzzy rules reflected quantum superposition principles. Rules prioritizing the internuclear direction revealed the system's 1D symmetry, aligning with Linear Combination of Atomic Orbitals theory--a novel data-driven perspective on orbital hybridization. Membership function variances (σ) further quantified electron delocalization trends, and peak prediction errors highlighted unresolved quantum cusps. The choice of functions critically impacted performance: Gaussian/Generalized Bell outperformed Sigmoid, with errors improving as training data increased, showing scalability. This study underscores the context-dependent value of ML: ANN for precision and ANFIS for interpretable, parameter-efficient approximations that link inputs to physical behavior. These findings advocate hybrid approaches in quantum simulations, balancing accuracy with explainability to accelerate discovery. Future work should extend ANFIS to multi-electron systems and integrate domain-specific constraints (e.g., kinetic energy terms), bridging data-driven models and fundamental physics.

Citations

• Recent Advances in Metasurface Design and Quantum Optics Applications with Machine Learning, Physics-Informed Neural Networks, and Topology Optimization Methods
• First principles physics-informed neural network for quantum wavefunctions and eigenvalue surfaces
• Solving the Schrödinger Equation in the Configuration Space with Generative Machine Learning
• Physics-Informed Neural Networks for Quantum Eigenvalue Problems
• ANFIS: adaptive-network-based fuzzy inference system

Related

• <i>Ab initio</i> solution of the many-electron Schrödinger equation with deep neural networks
• Quantum Machine Learning of Molecular Energies with Hybrid Quantum-Neural Wavefunction
• Adapting Quantum Machine Learning for Energy Dissociation of Bonds
• QINNs: Quantum-Informed Neural Networks
• Quantum machine learning interatomic potential: Application of variational quantum algorithm