Interpretable Activation-Selection Neural Networks for Symbolic Regression of Parameter-Dependent Hamiltonian Eigenvalues

AI summary

Overview

Research area: Quantum physics and machine learning, specifically interpretable symbolic regression for analytical approximations of Hamiltonian eigenvalues.

Technical level: Advanced. The paper assumes familiarity with parameter-dependent Hamiltonians, eigenvalue problems, degenerate perturbation theory, and neural-network-based symbolic regression.

Scope: The authors introduce and test an activation-selection network that converts trained neural-network weights into explicit analytical expressions for eigenvalues of effective spin-chain Hamiltonians relevant to zero-quantum nuclear magnetic resonance.

What This Paper Is About

Numerical diagonalization of parameter-dependent Hamiltonians produces eigenvalues but does not by itself reveal how those eigenvalues depend on the Hamiltonian parameters. The authors aim to obtain compact analytical approximations that expose that dependence. They propose a neural architecture that learns which simple mathematical functions to combine and then translates the trained network into a readable formula.

Key Contributions

1. An activation-selection network (ASN) for symbolic regression. The architecture is differentiable, and each input node learns a sparse combination of predefined analytical functions rather than using a fixed activation function. The trained network can be converted directly into an explicit expression.

2. Dimensionless-ratio preprocessing. Hamiltonian parameters and eigenvalues are expressed as dimensionless ratios before regression. This enforces dimensional homogeneity, reduces the number of independent variables, and makes the extracted expressions independent of the chosen energy units.

3. Polynomial generation through network depth. Using the function library {0, x, x²}, compositions across successive hidden layers produce polynomial expansions of progressively higher degree. The abstract states that polynomial expansions of arbitrary finite degree can in principle be obtained by increasing network depth.

4. A comparative evaluation against perturbation theory and fixed-basis least squares. The ASN is applied to effective three- and four-site spin-chain Hamiltonians relevant to zero-quantum nuclear magnetic resonance, and its expressions are compared with degenerate perturbation theory and with fixed-basis least-squares models.

Main Findings

• Perturbation-theory structure is recovered. Comparisons with degenerate perturbation theory show that the extracted expressions capture the expected constant, linear, and quadratic structure.

• Adaptive activation selection is not intrinsically more accurate. Fixed-basis least-squares models match or slightly outperform the ASN when an appropriate quadratic basis is specified in advance.

• A radial feature improves behavior near degeneracy. Including a radial feature improves the local approximation near the degeneracy.

• Unit-independent expressions are obtained. The dimensionless-ratio normalization ensures that the extracted expressions do not depend on the choice of energy units.

• Depth offers a route to higher-degree polynomials in principle. The abstract claims that arbitrary finite polynomial degree can be reached by increasing network depth; specific demonstrations beyond the tested cases are not reported in the abstract.

Methodology in Plain English

The authors first rewrite the Hamiltonian parameters and eigenvalues as dimensionless ratios, so the problem no longer depends on the units used to measure energy. They then build a neural network in which each input node chooses a sparse combination from a small library of functions, namely the constant, linear, and quadratic terms. Stacking hidden layers lets those simple pieces combine into higher-degree polynomials. Because the network is differentiable, it can be trained with standard gradient-based methods, and because its operations are explicit, the trained network can be read out as a formula rather than remaining a black box. The authors apply this procedure to effective spin-chain Hamiltonians with three and four sites that are relevant to zero-quantum nuclear magnetic resonance. They then compare the resulting formulas with expressions from degenerate perturbation theory and with least-squares fits that use a fixed basis chosen in advance.

Why This Matters

The work connects interpretable machine learning with analytical quantum physics. It offers a way to generate human-readable approximations when several functional forms seem plausible, while also giving a candid assessment of when such adaptive selection is actually useful compared with a well-chosen fixed basis.

Potential applications include:

• Magnetic resonance: interpreting and designing zero-quantum nuclear magnetic resonance experiments in spin-chain-like systems.
• Quantum hardware: building compact analytical models of parameter-dependent energy levels for calibration and control.
• Molecular and materials modeling: extracting interpretable parameter dependences where full diagonalization gives numbers but limited insight.
• Scientific software and education: producing explicit formulas that can be inspected, communicated, and reused outside a numerical solver.

Industry relevance: The approach is relevant to quantum-computing and spectroscopy communities that need interpretable, unit-independent models of energy spectra, and to developers of symbolic-regression tools who want physically constrained, dimensionally consistent outputs.

Future Directions

• Higher-degree and larger systems. Testing whether increasing network depth reliably produces higher-degree polynomial expansions for larger spin chains and more complex Hamiltonians.

• Broader function libraries. Exploring libraries beyond {0, x, x²}, including radial or other non-polynomial features, and determining when such features materially improve local approximations near degeneracies.

• Hybrid workflows. Using the ASN to identify a compact functional form and then fitting that form with fixed-basis least squares, since the abstract reports that a suitable predefined basis can match or slightly outperform the ASN.

• Systematic comparison with other symbolic-regression methods. Establishing when adaptive activation selection offers practical advantages over predefined bases across different Hamiltonian families and degeneracy structures.

Target Audience

Quantum physicists interested in analytical approximations to eigenvalue problems; magnetic resonance researchers working with spin-chain models and zero-quantum transitions; machine-learning researchers developing interpretable symbolic-regression architectures; and graduate students or practitioners who need compact, unit-independent formulas rather than purely numerical spectra.

Authors’ abstract

Analytical approximations to eigenvalues of parameter-dependent Hamiltonians can provide physical insight that is not readily apparent from numerical diagonalization alone. Here, we introduce an activation-selection network (ASN), a differentiable symbolic-regression architecture in which each input node learns a sparse combination of predefined analytical functions, and the trained network can be converted directly into an explicit expression. Before regression, the Hamiltonian parameters and eigenvalues are expressed as dimensionless ratios. This normalization enforces dimensional homogeneity, reduces the number of independent variables, and ensures that the extracted expressions do not depend on the choice of energy units. Using the library {0, x, x^2}, compositions across successive hidden layers generate polynomial expansions of progressively higher degree; polynomial expansions of arbitrary finite degree can therefore be obtained in principle by increasing the network depth. We apply the ASN to effective three- and four-site spin-chain Hamiltonians relevant to zero-quantum nuclear magnetic resonance. Comparisons with degenerate perturbation theory show that the extracted expressions capture the expected constant, linear, and quadratic structure. Fixed-basis least-squares models match or slightly outperform the ASN when an appropriate quadratic basis is specified in advance, while inclusion of a radial feature improves the local approximation near the degeneracy. These results establish the ASN as a differentiable framework for selecting compact symbolic representations when several functional forms are plausible, while showing that adaptive activation selection does not provide an intrinsic accuracy advantage over a suitable predefined basis.