Fuzzy Neural Network Performance and Interpretability of Quantum Wavefunction Probability Predictions

TL;DR

This work tackles predicting quantum wavefunction probability distributions, a task central to computational chemistry, by comparing Artificial Neural Networks (ANNs) and Adaptive Neuro-Fuzzy Inference Systems (ANFIS) using data generated by Physics-Informed Neural Networks (PINNs) for the H$_2^+$ ion. The authors demonstrate that ANNs achieve higher accuracy ($R^2 \approx 0.99$) but require an order of magnitude more parameters than ANFIS, which, while slightly less accurate ($R^2 \approx 0.95$ with Gaussian MFs), offers interpretable mappings where Gaussian membership functions encode proton-localized regions near equilibrium positions and fuzzy rules reflect quantum superposition and system symmetry. The work reveals that MF type (Gaussian/Generalized Bell vs Sigmoid) critically affects performance and that increasing training data improves ANFIS accuracy, suggesting potential for hybrid physics-guided models. Overall, the study advocates leveraging the complementary strengths of ANN precision and ANFIS interpretability to accelerate quantum simulations, with future work extending ANFIS to multi-electron systems and incorporating kinetic-energy terms to align more closely with fundamental quantum mechanics.

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 H$_{2}^+$ ion, leveraging data generated via Physics-Informed Neural Networks (PINNs). While ANN achieved superior accuracy (R$^2$ = 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.

Figures (8)

• Figure 1: Artificial Neural Network and Adaptive Neuro-Fuzzy Inference System architectures in (a) and (b), respectively.
• Figure 2: Wavefunction probability amplitude $|\Psi^2|$ of the PINN-generated solution for the H$_2^+$ ion, visualized as (a) a line profile along the $x$-axis (the internuclear direction) and (b) the two spatial dimensions, for the fixed plane $z=0$. In (a), the LCAO solution is presented as a reference comparison, lacking complex interaction terms between the atomic orbitals.
• Figure 3: Parity plots comparing the reference and predicted values for the (a) classical and (b) fuzzy methods, respectively, along with their corresponding models training and validation losses in (c) and (d).
• Figure 4: Comparison between the Gaussian, Generalized Bell, and Sigmoid membership function models; in (a-c) are presented the training and validation losses, and in (d-f), the parity plots of the predictions, respectively.
• Figure 5: Absolute errors for the probability amplitude in the $xy$-plane for the ANN, Gaussian, Generalized Bell, and Sigmoid models in (a-d), respectively.