ML-Driven Hamiltonian Reconstruction

• The paper demonstrates that machine learning models, including CNNs and MLPs, can invert observable data to accurately recover Hamiltonian parameters with sub-percent error.
• Machine-learning-driven Hamiltonian reconstruction is defined as the inference of effective system dynamics using supervised, unsupervised, and physics-informed approaches applied to real and synthetic measurements.
• The methodology enables high-throughput extraction of quantum, electronic, and magnetic interactions while addressing challenges such as noise robustness, dimensional scaling, and model bias.

Machine-learning-driven Hamiltonian reconstruction refers to the inference of the underlying Hamiltonian—i.e., the generator of dynamics or spectral structure—from experimentally accessible or simulated data using advanced machine learning methodologies. This paradigm encompasses supervised, unsupervised, and hybrid approaches, leveraging neural networks, statistical regressors, dimensionality reduction, and physically motivated priors to invert observation-to-Hamiltonian mappings in quantum, classical, and strongly correlated settings. Applications span single-particle and many-body systems, electronic structure, ultracold atoms, STM/IETS, ARPES, and beyond, facilitating direct comparison of model-based theory to experiment and enabling automated analysis of high-dimensional physical data.

1. Inverse Problems in Hamiltonian Reconstruction

Machine-learning-driven Hamiltonian reconstruction operationalizes the inverse mapping from a set of measurable descriptors (e.g., local density of states, dynamical correlators, image textures, time series, or many-body observables) to the parameters or functional form of the effective Hamiltonian. For example, in tight-binding models, the local density of states (LDOS) $\rho_E(r)$ at each position $r$ and energy $E$ can, in principle, uniquely determine the potential landscape $V(r)$ given sufficient resolution. More generally, in quantum magnets, the full many-body Hamiltonian, including isotropic and anisotropic exchange and Dzyaloshinskii–Moriya interactions, can be reconstructed from spatially resolved inelastic spectroscopy with engineered impurity configurations. In all these scenarios, the goal is to recover a functional form

$H[\theta] = \sum_j \theta_j h_j,$

where $\theta_j$ are parameters to be inferred and $\{h_j\}$ a fixed set of basis operators, using incomplete or noisy data. The central challenge arises due to ill-posedness: experimental observables are often only indirectly or non-invertibly related to $H[\theta]$, necessitating regularization, inductive biases, or data augmentation to ensure robust parameter recovery (Paul et al., 11 Sep 2025, Karjalainen et al., 21 Oct 2025, Basak et al., 2022).

2. Representative Machine Learning Architectures and Mapping Paradigms

Leading Hamiltonian reconstruction pipelines use supervised learning with convolutional or fully connected neural networks, unsupervised dimensionality reduction (e.g., autoencoders), and physics-informed models such as symplectic recurrent networks or analytic fitting via variational optimization.

• Supervised image-to-image regression: In the LDOS inversion framework, 1D and 2D convolutional neural networks (CNNs) are trained to map normalized $\tilde\rho(r)$ fields (obtained from LDOS maps) to normalized on-site potentials $\tilde V(r)$. These CNNs utilize architectures with multiple convolutional layers, residual blocks, and ReLU nonlinearities. Training minimizes mean squared error (MSE) between predicted and true potentials. Hyperparameters (kernel size, channels, depth, learning rate) are optimized with Bayesian methods such as Optuna (Paul et al., 11 Sep 2025).
• Fully connected networks with physically structured features: For impurity-tomography in quantum magnets, high-dimensional input data (concatenated spectra across sites, energies, and impurity placements) undergo principal component analysis (PCA) to reduce dimensionality. The reduced features are then fed to dense MLPs to regress normalized Hamiltonian parameters, exploiting ReLU activations and dropout for regularization (Karjalainen et al., 21 Oct 2025).
• Implicit neural representations: For ARPES data, sinusoidal activation-based networks (SIREN) are trained to interpolate the multi-dimensional mapping from momentum and Hamiltonian parameters to spectral intensities, enabling fast surrogate evaluation and differentiable end-to-end fitting (Zhang et al., 17 Mar 2026).
• Physics-informed recurrent and symplectic models: For dynamical systems, adaptable symplectic recurrent neural networks (ASRNNs) integrate Hamiltonian parameterization with symplectic time stepping (e.g., Störmer–Verlet), avoiding explicit derivative estimation and enabling training from extremely sparse, noisy temporal data (Thapar et al., 19 Apr 2026).
• Dimensionality reduction and feature selection: In systems with large operator bases, bottleneck autoencoders project local measurement vectors onto low-dimensional latent spaces, allowing candidate term selection via gradient ranking and iterative refinement (Nandy et al., 2023).

3. Data Acquisition, Synthetic Data Generation, and Input Representations

Training data for machine-learning-driven Hamiltonian reconstruction are primarily generated via simulation of physically motivated models, but can also be acquired from experiment under appropriately controlled conditions.

Method/Domain	Input Descriptor	Hamiltonian Model	Output
STM/LDOS inversion	$r$0 maps	Single-particle tight-binding	$r$1 image
Quantum magnets	STM-IETS/ESR spectra	Spin chain w/ Heisenberg + DMI	$r$2
Quantum materials	Image patches (SNIM, ARPES)	Ising/RFIM or TB models	Hamiltonian class/parameters
Dynamical systems	Sparse time series $r$3	Nonlinear ODE Hamiltonian	Symbolic $r$4
Molecular systems	Electron densities/time series	Linear Fock/TDHF Hamiltonian	Hamiltonian matrix

Synthetic data are typically generated by: (i) random sampling of potential or interaction landscapes from Gaussian fields or constrained distributions; (ii) diagonalization of the resulting Hamiltonian to produce “forward” observable data (LDOS, spectra, or dynamics); (iii) adding realistic noise, both uncorrelated (additive Gaussian) and correlated (spatial or temporal correlations matching experimental conditions). Normalization of observables and regression targets ($r$5-scoring) ensures network stability and faster convergence (Paul et al., 11 Sep 2025, Karjalainen et al., 21 Oct 2025, Basak et al., 2022, Thapar et al., 19 Apr 2026).

4. Performance Metrics, Robustness, and Generalization

The fidelity and reliability of Hamiltonian inference are quantified by diverse performance metrics:

• Mean squared error (MSE): Core metric for regression between predicted and true potentials or parameter vectors. In LDOS inversion, CNNs reduce MSE to $r$6–$r$7, outperforming 1NN/10NN baselines by more than an order of magnitude (Paul et al., 11 Sep 2025).
• Pearson-style correlation/fidelity: Used for parameter estimation in quantum magnets; networks achieve $r$8 for isotropic, anisotropic, and third-neighbor exchange terms at zero noise, degrading only for DMI (which is less visible without off-diagonal observables) (Karjalainen et al., 21 Oct 2025).
• Noise robustness: Models are typically robust to moderate additive/correlated noise, with performance sustained up to signal-to-noise ratios near unity, especially after retraining on noised data (Paul et al., 11 Sep 2025). Denosing autoencoders in continuous quantum measurement inversion demonstrate systematic learning curves with increasing data volume (Tucker et al., 2024).
• Out-of-distribution and parameter drift: CNNs for LDOS inversion maintain low MSE across moderate shifts in the disorder amplitude $r$9 and correlation length $E$0. Implicit neural rep SIREN surrogates generalize across distinct material classes (nickelates, manganites) without retraining (Paul et al., 11 Sep 2025, Zhang et al., 17 Mar 2026).
• Sample and computational complexity: For methods employing local correlators or “snapshots,” the inference cost is amortized and remains orders of magnitude below Bayesian or grid-search approaches, with prediction times on the order of milliseconds once models are trained (Valenti et al., 2021).

5. Application Domains and Experimental Relevance

Machine-learning-driven Hamiltonian reconstruction has been deployed across a range of platforms:

• Scanning tunneling microscopy (STM): Experimental dI/dV maps, proportional to LDOS at a given $E$1, are directly invertible to $E$2 via trained CNNs, enabling extraction of local energy landscapes in real materials. The approach is robust to thermal broadening, instrumental noise, and tip convolution through retraining on appropriate noise models (Paul et al., 11 Sep 2025). Advanced applications include setpoint-dependent STM-IETS, where full spectral images as a function of tip–sample distance are mapped to multiorbital Hamiltonian parameters using supervised feedforward networks (Lupi et al., 27 Jan 2026).
• Quantum magnetism and impurity tomography: The mapping from spatially resolved excitation spectra under engineered impurity placement to long-range and competing interaction terms in designer quantum magnets is realized via supervised MLPs trained on concatenated multi-geometry spectra. This protocol recovers exchange and DMI terms with sub-percent accuracy for the majority of parameters, providing a route for bottom-up quantum simulation validation (Karjalainen et al., 21 Oct 2025).
• Structural and disordered materials: High-entropy alloy modeling is performed by fitting local energies from linear-scaling DFT to cluster expansion or deep neural architectures, enabling recovery of effective pair and multi-site interactions with $E$3 (Liu et al., 2019). Ising motif-based approaches reduce phase diagram reconstruction and disordered parameter estimation to unsupervised histogram clustering and statistical distance minimization (Valleti et al., 2020).
• AR/strings in quantum materials: Automated extraction of minimal tight-binding parameters $E$4 from 4D ARPES data via implicit neural surrogates yields parameter uncertainty below 2%, with direct Fermi-surface and dispersion matching to experimental observations (Zhang et al., 17 Mar 2026).

6. Methodological Limitations and Pathways to Generalization

Current frameworks are subject to several practical and conceptual limits:

• Feature completeness and prior bias: Supervised pipelines reconstruct the Hamiltonian only within the space defined by the simulated training data and architectural inductive biases. Unseen physical processes, higher-order interactions, or experimental artifacts not captured in the forward simulations can result in model misspecification or inference bias (Paul et al., 11 Sep 2025, Lupi et al., 27 Jan 2026).
• Noise and cross-channel limitations: For parameters such as DMI that couple only weakly or diagonally to available dynamical observables, the absence of cross-channel measurement data (e.g., off-diagonal correlators) can fundamentally constrain inference resolution (Karjalainen et al., 21 Oct 2025).
• Dimensional exaggeration and scaling: While pointwise and low-dimensional regression metrics show excellent performance, the curse of dimensionality and exponential operator basis expansion limit direct application to higher-dimensional, strongly correlated bulk systems. Compression techniques (PCA, autoencoders, harmonic expansions) and locality-reduction heuristics are essential for tractability (Nandy et al., 2023, Khosravian et al., 2023).
• Experimental translation and retraining: To translate trained models to experiment, retraining with augmented synthetic data reflecting specific noise, disorder, and instrumental limitations is required. In STM and ARPES, system-specific simulated data and multi-channel observables enhance real-world deployment (Paul et al., 11 Sep 2025, Zhang et al., 17 Mar 2026).

Pathways for extension include Bayesian or uncertainty-quantified regression, incorporation of additional spatial/energy channels, active learner protocols for experiment-driven data collection, and integration of higher-order correlation functions (Karjalainen et al., 21 Oct 2025).

7. Impact and Outlook

Machine-learning-driven Hamiltonian reconstruction has demonstrated pronounced impact on automated, high-throughput analysis of experimental material data, tight-binding model extraction, and many-body simulation validation. Performance benchmarks—including sub-percent-accuracy parameter retrieval, robustness to realistic noise, and orders-of-magnitude acceleration over conventional grid-search approaches—have been established in both simulated settings and direct application to quantum material experiments (Paul et al., 11 Sep 2025, Karjalainen et al., 21 Oct 2025, Zhang et al., 17 Mar 2026, Valenti et al., 2021). Methods are being generalized to accommodate more complex (multi-orbital, multi-band, topologically nontrivial, or disordered) Hamiltonians, and to enable direct, interpretable, and scalable inference of physical models from high-dimensional observation data. As methodological innovations advance in neural architectures, feature engineering, and robust training under experimental nonideality, further integration with experimental quantum platforms and real-time materials design pipelines is anticipated.