Machine learning and the physical sciences

Giuseppe Carleo; Ignacio Cirac; Kyle Cranmer; Laurent Daudet; Maria Schuld; Naftali Tishby; Leslie Vogt-Maranto; Lenka Zdeborová

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Machine learning and the physical sciences

Giuseppe Carleo, Ignacio Cirac, Kyle Cranmer, Laurent Daudet, Maria Schuld, Naftali Tishby, Leslie Vogt-Maranto, Lenka Zdeborová

arXiv:1903.10563v2Mar 25, 2019physics.comp-phastro-ph.COcond-mat.dis-nnhep-thquant-ph

TL;DR

Machine learning and the physical sciences have developed along intersecting but distinct paths: ML extracts patterns from data, whereas physics seeks mechanisms. This review synthesizes physical insights into ML, ML applications across physics, and the resulting methodological successes and challenges.

Problem
The interface between machine learning and physical sciences spans diverse methods and domains, requiring a selective account of their shared techniques, differing goals, applications, and challenges.
Method
The paper reviews conceptual developments, physics applications, and cross-fertilization across statistical, high-energy, quantum, cosmological, chemical, and materials physics.
Results
The review finds that machine-learning research in physical sciences is moving from exploratory toy models toward real experimental data and stronger understanding of theoretical limitations.
Takeaways & Limitations
The interaction between machine learning and physical sciences remains in its infancy, with further results anticipated from continued interplay between the fields.

Abstract

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Machine learning encompasses a broad range of algorithms and modeling tools used for a vast array of data processing tasks, which has entered most scientific disciplines in recent years. We review in a selective way the recent research on the interface between machine learning and physical sciences. This includes conceptual developments in machine learning (ML) motivated by physical insights, applications of machine learning techniques to several domains in physics, and cross-fertilization between the two fields. After giving basic notion of machine learning methods and principles, we describe examples of how statistical physics is used to understand methods in ML. We then move to describe applications of ML methods in particle physics and cosmology, quantum many body physics, quantum computing, and chemical and material physics. We also highlight research and development into novel computing architectures aimed at accelerating ML. In each of the sections we describe recent successes as well as domain-specific methodology and challenges.

I. INTRODUCTION

Machine learning and physics share data-driven modeling goals but differ in their relationship to mechanisms and interpretability. This review surveys their intersections, from statistical-physics insights into ML to applications across physics and related sciences.

I. INTRODUCTION: ML recognizes patterns in data to inform decisions about unseen problems, with recent advances enabling applications such as image recognition and game playing.The review presents ML as a broad set of data-processing tools whose recent successes expanded its use across industry and science.
I. INTRODUCTION: Physics and ML both gather data and build predictive models for complex systems, but physics emphasizes mechanisms while ML extracts patterns from data using comparatively opaque models.This difference motivates both enthusiasm about ML tools in physics and concern about their interpretability.
I. INTRODUCTION: The review selectively covers conceptual developments motivated by physics, ML applications in physics, and cross-fertilization across statistical, quantum, high-energy, and cosmological domains.It also discusses domain-specific challenges and novel computing architectures aimed at accelerating ML.
I. INTRODUCTION: Supervised learning uses labeled examples to learn predictions for unseen inputs, with training and test sets separating fitting from performance evaluation.Classification commonly uses accuracy, whereas regression commonly uses mean-squared error.
I. INTRODUCTION: Training adjusts model weights to minimize empirical risk, commonly through gradient descent or stochastic gradient descent, while generalization error measures performance on unseen data.Learning rate, initialization, and regularization can affect practical performance.
I. INTRODUCTION: Linear regression and regularized ridge regression provide basic supervised-learning methods, while kernel ridge regression and support-vector methods extend this toolkit, especially for limited samples.Ridge regression is described as important when dimensionality is high and sample size is not much larger than dimension.

2. Unsupervised learning and generative modelling

Unsupervised and generative methods seek structure or distributions in unlabeled data, while reinforcement learning learns actions through interaction and reward. The section highlights physically interpretable models, bottleneck representations, and successful deep reinforcement-learning applications.

2. Unsupervised learning and generative modelling: Unsupervised learning uses unlabeled inputs to recover underlying structure, including through clustering and generative modeling.Generative models seek distributions that produce samples statistically similar to observed data and may use latent variables.
2. Unsupervised learning and generative modelling: Boltzmann machines provide a physically appealing unsupervised-learning method because they can be interpreted as inverse Ising models.The review connects this interpretation to applications of Boltzmann and restricted Boltzmann machines across physics.
2. Unsupervised learning and generative modelling: Autoencoders reproduce inputs through a narrow bottleneck to learn succinct representations retaining salient features, while variational autoencoders add variational inference for unsupervised generative modeling.This approach reuses methods developed for supervised learning in an unsupervised setting.
2. Unsupervised learning and generative modelling: GANs use generator and discriminator networks to support unsupervised learning, while normalizing flows and autoregressive models offer tractable likelihoods for maximum-likelihood training.Semisupervised and active learning combine supervised and unsupervised elements when labels are partial or acquirable.
3. Reinforcement learning: Reinforcement learning trains an agent to choose actions that maximize reward through interaction with an environment, balancing exploitation and exploration.Deep Q-learning represents large value functions with neural networks when a complete Q matrix cannot be stored.
3. Reinforcement learning: AlphaGo and AlphaGo Zero achieved super-human performance in Go, illustrating a prominent success of reinforcement learning.Robot locomotion is another cited application.

II. STATISTICAL PHYSICS

Statistical physics has long contributed analytical tools for understanding learning, while modern ML creates new theoretical puzzles involving generalization, tractability, and architecture choice. Spin-glass methods and phase-transition analyses provide results for low-rank estimation and community detection, although rigor and algorithmic limits remain important concerns.

II. STATISTICAL PHYSICS: Statistical physicists helped establish learning theory and neural-network analysis through concepts including associative memory, spin-glass theory, and learning curves.These contributions predate the recent widespread use of ML in physics.
II. STATISTICAL PHYSICS: Deep neural networks expanded learning problems to millions of input dimensions, larger datasets, and many adjustable parameters, while achieving strong pattern-recognition performance with SGD.The scale shift helped drive renewed interest in theoretical explanations.
II. STATISTICAL PHYSICS: Worst-case PAC-like theory does not explain why highly overparameterized deep networks can generalize well on true labels.The observed behavior contrasts with the absence of useful bounds in the relevant regime.
II. STATISTICAL PHYSICS: Open theoretical questions include which learning problems are computationally tractable and how hyperparameters and architectures should be chosen beyond trial and error.Many learning problems are NP-hard in the worst case, according to the review.
II. STATISTICAL PHYSICS: The review is selective and excludes substantial statistical-physics work on recurrent neural networks with biological applications.Its scope focuses on recent physics contributions judged to have important impact on current learning theory and ML.
II. STATISTICAL PHYSICS: Low-rank decomposition models noisy data through a rank-r structure with r much smaller than dimensionality and sample count, with teacher-student analysis targeting latent-variable recovery.Spin-glass theory can compute information-theoretically optimal estimation error in an asymptotic limit, but early work lacked rigor and algorithmic-tractability understanding.
II. STATISTICAL PHYSICS: For stochastic block models, spin-glass analysis identified asymptotically optimal performance, sharp parameter regions for belief propagation, and phase transitions separating impossible, difficult, and efficient clustering.These results also motivated new spectral algorithms for sparse data and attracted follow-up work across several fields.
II. STATISTICAL PHYSICS: Recent progress on approximate message passing and related analyses has helped move statistical-physics research on algorithmic limitations into mainstream statistics and machine learning.The review presents this as part of a broader evolution from toy models toward experimental data and theoretically justified applications.

2. Restricted Boltzmann machines

Restricted Boltzmann machines connect unsupervised learning with statistical physics through their inverse-Ising interpretation. Physics-based analyses address their representations and training dynamics, alongside related generative approaches including VAEs, GANs, and autoregressive models.

2. Restricted Boltzmann machines: Boltzmann machines, especially restricted Boltzmann machines, are unsupervised-learning methods closely related to statistical physics and often called inverse Ising models.This connection has motivated extensive physics study and applications across physics domains.
2. Restricted Boltzmann machines: Random sparse restricted Boltzmann machines can represent compositional structure even with a single layer for a range of hidden-unit potentials.This result concerns models whose weights are imposed to be random and sparse rather than learned.
2. Restricted Boltzmann machines: Analytically studying RBM learning is difficult because contrastive divergence commonly relies on Gibbs sampling, though early learning dynamics can be linearized.Statistical-physics work also explores replacing Gibbs sampling with Thouless-Anderson-Palmer equations.
2. Restricted Boltzmann machines: Variational autoencoders connect latent-variable priors and variational inference with autoencoding, while single-hidden-layer VAEs relate to dictionary learning and sparse coding.These methods provide a physics-friendly perspective on unsupervised and generative learning.
2. Restricted Boltzmann machines: GANs generate samples resembling a training set, and physics-inspired solvable models connect GAN analysis to statistical-physics work on online perceptron learning.The cited work frames GANs as a generalization of earlier statistical-physics approaches.
2. Restricted Boltzmann machines: Autoregressive generative models are explicit probabilistic models that permit direct unbiased sampling and have been applied to statistical and quantum physics.Their appeal in this context stems from the availability of direct sampling.

1. Perceptron and GLMs

Statistical physics provides analytical tools for understanding generalization in high-dimensional and deep learning models, while key questions remain open for realistic settings.

Perceptron and GLMs: High-dimensional GLMs remain difficult to analyze when sample and parameter dimensions grow together, leaving optimal generalization behavior unresolved.The relevant regime fixes n/p = α while both n and p become large.
Perceptron and GLMs: Teacher-student models and replica methods connect mutual information to free energy and yield optimal estimation and generalization results.The teacher generates labels from a ground-truth weight vector and an output channel, while the student infers the rule.
Perceptron and GLMs: Committee machines exhibit a specialization transition: hidden units share weights at low sample counts and differentiate only above a threshold.The analyzed limit has k = O(1), n/p = α = O(1), and admits closed-form learning errors.
Perceptron and GLMs: Information-bottleneck theory characterizes deep-learning representations as retaining label-relevant information while discarding unnecessary input information.The theory replaces conventional capacity measures with the exponent of mutual information in the compressed representation.
Perceptron and GLMs: Statistical-physics analyses link SGD-driven representation compression and neural-network learning to diffusion, glassy landscapes, and wide low-loss regions.These works propose mechanisms involving irrelevant-dimension diffusion, reduced glassiness under over-parameterization, and attraction toward wide minima.
Perceptron and GLMs: Direct mutual-information calculations remain difficult in large networks, and current theories often rely on simplified models rather than realistic data distributions.Scalable approximations exist for linear or approximately rotationally invariant networks, but realistic universality classes remain a challenge.

E. Applications of ML in Statistical Physics

Machine learning is being applied across statistical physics to discover representations, identify phases, model dynamics, and explore new renormalization schemes, while realistic generalization remains unresolved.

Applications of ML in Statistical Physics: Neural networks have been proposed to learn renormalization schemes, but their ability to produce new physical discoveries remains unestablished.The open question concerns models that were not previously well understood.
Applications of ML in Statistical Physics: Neural networks can learn order parameters and locate phase transitions without prior specification of the relevant physical order parameter.This approach has been explored in Ising, XY, and frustrated magnetic systems.
Applications of ML in Statistical Physics: The reliability of learned phase identifiers remains unclear, especially for distinguishing true thermodynamic transitions from crossovers.The limitations of identifying previously unknown order parameters also remain to be clarified.
Applications of ML in Statistical Physics: Multi-scale clustering and learning methods have been used to identify structural, dynamical, and history-dependent features in disordered and glassy solids.Examples include spatial structures, flow defects, and parameters capturing history dependence.
Applications of ML in Statistical Physics: Reservoir computers predict chaotic and weather-model trajectories, while reinforcement learning has taught an autonomous glider to soar.These applications illustrate ML tools for studying, predicting, and controlling nonlinear dynamical systems.
Applications of ML in Statistical Physics: Statistical-physics theories offer closed-form results through toy generative models, but they must move beyond iid inputs toward models resembling practical neural-network data.The review identifies realistic universality classes as a central challenge and says ML’s full potential in these systems remains uncovered.

A. The role of the simulation

Simulation is central to machine learning in particle physics and cosmology because it supplies labeled data, while current research seeks methods that address simulation cost, uncertainty, and mismatch with real data.

The role of the simulation: High-fidelity simulations generate labeled training data either by conditioning directly on targets or by computing labels from latent simulation variables.The resulting pairs are used for supervised learning in particle physics and cosmology.
The role of the simulation: Particle-physics and cosmological simulations differ in structure, with stochastic collider interactions contrasting with deterministic cosmic evolution from stochastic initial conditions.Cosmological N-body simulations are expensive and relatively few, whereas collider simulations require enormous numbers of collisions.
The role of the simulation: Simulation-based ML research targets data efficiency, uncertainty incorporation, weak supervision, anomaly detection, simulation tuning, and fast surrogate models.These directions include approximate inference and neural surrogates for both synthetic-data generation and statistical inference.
The role of the simulation: Machine learning supports particle identification, reconstruction, and event selection using detector inputs and labels such as particle species, energy, or momentum.These models process large experimental data streams, including LHC data.
The role of the simulation: Deep learning improved event selection over boosted decision trees in an early proof-of-concept without requiring engineered features.The demonstration used a deep multilayer perceptron trained on a very large simulated dataset and simplified detector setup.

1. Jet Physics

Jet and neutrino analyses use convolutional and graph-based models for structured detector data, while simulation mismatch motivates robust, weakly supervised, and data-driven approaches.

Jet Physics: Boosted and fat-jet classification targets overlapping jet substructure from beyond-Standard-Model decays against abundant quark- and gluon-initiated backgrounds.Jet characterization also supports theoretically robust tests of quantum chromodynamics.
Jet Physics: Gaussian processes model uncertain smooth spectra and alternative signal hypotheses as intensities of inhomogeneous Poisson point processes.The framework applies to particle physics, astrophysics, and cosmology.
Jet Physics: 40% higher electron-neutrino selection efficiency at the same purity was achieved by NOνA’s deep convolutional network.The network simultaneously processed two images from different detector vantage points.
Jet Physics: Graph neural networks for IceCube signal classification outperformed both a traditional physics-based method and a classical 3D convolutional network.The detector array was represented as a graph whose vertices are sensors and whose edges depend on sensor coordinates.
Robustness to systematic uncertainties: Simulation imperfections motivate uncertainty propagation, domain adaptation, and strategies that avoid modeling p(X|y) directly with simulation.These approaches address covariate shift and systematic mismodeling between training and target data.
Robustness to systematic uncertainties: Weakly supervised methods use regions with known class proportions or assumptions about relative probabilities, but their distribution-independence assumption is context-dependent.The approach has been applied to train jet taggers without fully specified labels.
Jet Physics: Approximately 70% of data selected by the LHC trigger is selected by machine-learning algorithms, with specialized hardware supporting real-time deployment.Tools compile ML models for field-programmable gates arrays used in triggering systems.

5. Theoretical particle physics

Machine learning is applied across theoretical and observational problems in particle physics and cosmology, including parameter inference, lens finding, and simulation-based analysis. These applications use neural networks and related methods while facing data-representation, simulation, and distribution-shift challenges.

Theoretical and computational applications: Machine learning characterizes theoretical landscapes and supports simulation-focused computing architectures for physics applications.Applications include string-theory landscape characterization and systems combining high-performance computing with machine learning.
Photometric redshift: Photometric redshift estimation uses spectroscopic measurements as ground truth for regression from broadband photometry or imaging.Traditional template fitting and machine-learning methods, including neural networks and boosted decision trees, have both been used.
Photometric redshift: Training-data covariate shift from spectroscopic selection effects and photometric-survey subtleties requires critical validation.The Dark Energy Survey established a validation process to evaluate approaches under these distributional concerns.
Gravitational lens finding and parameter estimation: Neural networks classify rare strong gravitational lenses and can support computationally intensive nonlinear lens-parameter estimation.Training data are generated with PICS lensing ray-tracing and LensPop mock LSST observations.
Other examples: 3D convolutional networks predict cosmological parameters from dark-matter spatial distributions generated by N-body simulations.The proof-of-concept assumes specific values for 10 parameters in the standard ΛCDM model; visible-matter applications would require modeling tracer bias and variance.
Other examples: Higher-order or richer weak-lensing representations can break degeneracies between modified-gravity models with massive neutrinos and standard ΛCDM predictions.A wavelet representation supplied to a convolutional network was used to discriminate between models.

D. Inverse Problems and Likelihood-free inference

Inverse problems in physical sciences are often unstable or involve simulator-defined likelihoods that cannot be evaluated analytically. Likelihood-free methods address this by using forward simulation and learned surrogates, while introducing challenges in high-dimensional inference.

Inverse problems: Inverse problems can be ill-posed, so small changes in observations may produce large changes in estimates and high estimator variance.Poor conditioning contributes to instability when the forward model is a linear operator.
Likelihood-free inference: Simulator-defined likelihoods are often intractable because integrating over enormous latent spaces is analytically impossible and difficult to estimate in high dimensions.Monte Carlo samples can be generated, but histogram methods do not scale well and kernel density estimation is reliable only to around 5 dimensions.
Likelihood-free inference: Likelihood-free inference uses forward simulation without requiring direct evaluation of the simulator’s likelihood.This approach was motivated by the foundational role of intractable simulator likelihoods across particle physics, cosmology, epidemiology, and phylogenetics.
Approximate Bayesian Computation: Approximate Bayesian Computation accepts simulations according to a distance threshold, recovering exact Bayesian inference as ϵ → 0 while losing efficiency.For high-dimensional observations, choosing a distance measure that preserves acceptance efficiency and inference quality is difficult.
Neural likelihood-free methods: Neural likelihood-free methods learn approximate likelihoods or likelihood ratios using density estimators, normalizing flows, and simulation-trained neural networks.In effective-field-theory applications, joint likelihood ratios and joint scores can be tractable even when ordinary likelihoods are not.
Extensions: Likelihood-free inference also supports probabilistic-programming inference over latent variables and adversarial or variational optimization of simulation parameters.These approaches include tau-lepton decay inference and simulation tuning through differentiable surrogate losses.

3. Examples in Cosmology

Cosmological machine learning uses likelihood-free inference, generative modeling, and domain-aware representations to analyze observations and accelerate simulation-related tasks. Applications include galaxy-image calibration, structure formation, and broader survey and experiment strategies.

Likelihood-free inference: Cosmological applications of Approximate Bayesian Computation include constraining Milky Way thick-disk formation and galaxy morphological-transformation rates.These applications motivated tools such as CosmoABC for streamlining cosmological use of the methodology.
Likelihood-free inference: Learned summary statistics can compress high-dimensional observations while maximizing Fisher information about cosmological parameters.The summaries approximate sufficient statistics for the implicit likelihood in a small neighborhood of parameter space.
Generative modeling: Generative modeling methods include GANs, VAEs, autoregressive models, and normalizing flows for learning distributions resembling empirical data.These models are relevant when likelihoods are unknown or computationally intractable.
Generative modeling: Traditional scientific simulators provide physically grounded causal data-generation models but are often slow because distributions emerge from low-level microphysical descriptions.Examples include LHC collision simulation and cosmological simulation tasks.
Generative modeling: Generative models have been used for particle calorimeter deposits and cosmological structure formation, including hybrid neural-network approaches.These applications aim to learn or approximate simulation outputs for physics workflows.
Galaxy-image calibration: GANs enable an implicit generalization of the parametric bootstrap for calibrating galaxy-shape measurements in weak-lensing surveys.The calibration problem requires large collections of high-quality galaxy images that are expensive to collect.
Domain-aware methods: Machine learning is positioned as a key strategy for upgraded high-luminosity collider programs and future cosmology and particle-physics experiments.A parallel research direction incorporates hierarchical, compositional, geometrical, and symmetry-based domain knowledge into models.

A. Neural-Network quantum states

Neural-network quantum states represent many-body wave-functions with artificial neural networks and support supervised, unsupervised, and variational quantum applications. Their capabilities are demonstrated across several systems, but expressive power, fermionic symmetries, and learning difficulty remain open challenges.

Representations: Neural-network quantum states parameterize many-body wave-functions using artificial neural networks, including feed-forward networks and restricted Boltzmann machines.RBM states can use complex-valued weights and have been extended to deeper and convolutional architectures.
Challenges: Imposing physical symmetries is a central challenge, especially fermionic antisymmetry, which requires the wave-function to change sign when particle positions are exchanged.Convolutional architectures can impose spatial symmetries, whereas more involved quantum symmetries require redesigned architectures.
Challenges: The representational relationship between NQS and computationally efficient tensor networks remains open, while compact RBM states cannot represent all physical states in general.Deep Boltzmann Machines are introduced to address intrinsic limitations of shallow RBM architectures.
Learning difficulty: Learning some random matrix-product states is hard, but the source of this hardness—optimization landscapes versus limitations of shallow NQS—remains unclear.The suspected connection is to the entanglement structure of the random states.
Learning tasks: NQS studies address supervised learning of target states, unsupervised reconstruction from measurements, and variational approximations to quantum ground states.Quantum state tomography requires measurements from multiple bases to reconstruct complex phases.
Applications: Experiments on spin, bosonic, and fermionic models obtain results competitive with existing state-of-the-art approaches, with improvements reported for selected two-dimensional lattice and topological systems.Variational NQS methods also address time-dependent Schrödinger and open dissipative systems.

B. Speed up many-body simulations

Machine learning is applied to many-body simulations and data analysis, including sampling, spectral reconstruction, phase classification, and physics-inspired tensor-network models. These approaches show progress on selected problems but face important limitations for non-local phases and experimentally grounded discovery.

Simulation and inference: Machine learning is explored as a way to reduce or bypass the sign problem affecting Quantum Monte Carlo for fermionic and frustrated spin systems.ML is also used for analytical continuation of imaginary-time correlations as an alternative to maximum-entropy methods.
Synthetic data: Many-body localization provides a benchmark where ML can effectively learn phase transitions in relatively small systems accessible to exact diagonalization.Methods using experimentally relevant local dynamics appear most promising for experimental applications.
Synthetic data: Topological phases remain difficult because their non-local order parameters are hard for standard image-classification schemes to learn from raw configurations.Pre-engineered features can help, but they rely on assumptions about the phase transition and may reduce effectiveness for discovering new phases.
Experimental data: Supervised learning has classified experimental quantum data, including phases in ultracold atoms and competing theories for Hubbard-model snapshots.The Hubbard-model results were reported as compatible with one proposed theory with good confidence.
Physics-inspired ML: Tensor-network representations such as matrix product states and tree tensor networks provide physics-inspired alternatives for classification, time-series modeling, and generative modeling.Tree tensor networks have been applied to classification and generative tasks with reported success.

E. Outlook and Challenges

Machine learning has rapidly expanded across quantum many-body physics and quantum computing, with improved performance on selected problems and promising applications in quantum-state processing. The review emphasizes that the field remains incomplete, with unresolved representation, experimental, and implementation challenges.

Outlook: Machine learning applications to quantum many-body problems span numerical simulation and data analysis, with improved performance over existing techniques on selected problems.The review states that the broader potential of ML in this domain has only been partially demonstrated.
Many-body challenges: The origins of empirical success in neural-network quantum states remain less understood than for tensor networks, and efficient representations of fermionic systems are still needed.These issues remain key open challenges in variational many-body studies.
Cross-fertilization: Tensor-network and complex-valued architectures help connect physics tools with computer science, while future work must preserve the generality of physics-based methods.The review identifies effective interfacing with the computer-science community as a challenge.
Experimental outlook: Experimental-data applications in quantum many-body physics remain in their infancy compared with mature ML use in high-energy and astrophysics.Progress toward standard data-analysis tools requires closer collaboration between theoretical and experimental efforts.
Quantum computing: In quantum computing, ML is used for quantum-state tomography and benchmarking, quantum-state preparation and control, and quantum error correction.The review also covers applications including quantum-circuit simulation, algorithm design, automated experiments, and measurement-based information extraction.
Quantum computing: ML has shown promising results across these quantum-computing problems and may become part of the toolbox alongside established methods.This is presented as a longer-run outlook rather than a completed transition.

A. Quantum state tomography

Quantum state tomography uses machine learning to reduce measurement demands and reconstruct quantum states, while its scalability remains constrained by intrinsic complexity and realistic noise.

A. Quantum state tomography: Full quantum state tomography reconstructs an unknown density matrix but requires a number of measurements that scales exponentially with system size.The task is central for assessing quantum-information platforms.
A. Quantum state tomography: Machine-learning and compressed-sensing approaches reduce measurement requirements by exploiting structure in the target density matrix.Compressed sensing reduces the measurement count from d2 to O(rd log(d)2) for rank-r states.
A. Quantum state tomography: ML approaches cannot improve the intrinsic exponential scaling of full quantum state tomography without assumptions about quantum-state regularity.Scalability to larger systems and noisy syndrome measurements remain challenges in related quantum-learning applications.
A. Quantum state tomography: Parameterization-based ML tomography has demonstrated reconstruction of highly entangled states up to about 100 qubits, beyond full-tomography techniques.The approach assumes regularity properties and can use neural quantum states for unsupervised learning.
B. Controlling and preparing qubits: Reinforcement learning is competitive for quantum control when interventions form a naturally discretized, sufficiently small action space.Applications include state preparation in interacting-qubit systems and Floquet-engineered states.
B. Controlling and preparing qubits: Bayesian optimization found a condensate-cooling protocol using 10 times fewer iterations than pure optimization techniques.A Gaussian process modeled the relationship between control parameters and condensate quality.

A. Energies and forces based on atomic environments

Machine learning models molecular and materials energies, forces, free-energy surfaces, and electron densities using representations designed around atomic environments and physical structure.

A. Energies and forces based on atomic environments: Neural networks predict molecular energies across isomers and non-equilibrium configurations using many-body symmetry functions describing local atomic neighborhoods.Atom-wise energy decompositions use separate neural networks for each element.
A. Energies and forces based on atomic environments: Adding layers that model interactions between atom-centered neural networks improved molecular energy predictions.This extends the atom-wise representation by allowing cross-atomic interactions.
A. Energies and forces based on atomic environments: Smooth overlap of atomic positions kernels provide continuous, differentiable representations suitable for molecular reactions and materials phase transformations.The representations also encode translation- and rotation-invariant atomic environments.
B. Potential and free energy surfaces: Neural networks can learn free-energy surfaces as functions of collective variables, supporting sampling and prediction of ensemble-dependent observables.Applications include isothermal compressibility of solid xenon and NMR spin-spin J couplings of a peptide.
B. Potential and free energy surfaces: Boltzmann generators sample equilibrium distributions in collective-variable spaces and provide states representing distributions on free-energy surfaces.They support analysis of transitions between minima and conformational basins.
C. Materials properties: ML interatomic potentials predicted amorphous-silicon structure and material properties using DFT training data from only small simulation cells.Accurate potentials for phase changes and defect energies across atomistic and material scales remain difficult.
D. Electron densities for density functional theory: ML can generate electron densities from nuclear potentials and use electron densities to learn molecular energies for reactive proton-transfer simulations.The nuclear potentials may be represented as sums of Gaussian functions.
D. Electron densities for density functional theory: Physics-informed ML functionals achieve only partial success when multiple physical constraints are imposed, and accurate derivatives remain an open question.The effect of constraints such as scaling laws and size-consistency is system-dependent.

E. Data set generation

Chemical and materials ML depends on carefully designed datasets, descriptors, and sampling strategies, while broader physics-inspired computing architectures offer possible acceleration routes.

E. Data set generation: QM9 contains 134,000 molecules, while COMP6 combines randomly sampled subsets of small-molecule and peptide datasets using a common optimization method.These datasets support standardized comparisons between quantum-chemistry ML methods.
E. Data set generation: Training structures should convey conformer changes, and dense sampling of chemical-composition space is not always necessary.Active learning and molecular-dynamics-based generation can add informative configurations efficiently.
E. Data set generation: Kernel-PCA descriptors cluster thousands of ice structures that differ by proton disorder or stacking faults and identify crystalline ice phases.Clustering groups structurally similar candidate materials.
E. Data set generation: Neural networks may require prohibitively expensive training data, while Gaussian processes were more accurate and used fewer points for a formaldehyde vibrational-frequency task.Gaussian-process points require more careful selection.
F. Outlook and Challenges: Future physics-based ML should incorporate symmetries, boundary behaviors, scaling laws, and uncertainty from calculations and measurements.These directions aim to improve training efficiency, regularization, and model transferability.
F. Outlook and Challenges: Optics, nanophotonics, and quantum computers are being investigated as novel AI accelerators.Optical processing offers speed, massive parallelism, and low power consumption.

C. Revealing features in data

Physics-based hardware can preprocess data, implement quantum learning methods, and accelerate machine-learning computations. These approaches range from optical random embeddings to quantum kernels, while current hardware and theoretical limitations constrain their maturity.

Optical and quantum hardware: Optical components can emulate neural-network layers, while quantum optics can encode information into light and turn the network into a universal quantum computer.The layer-wise map is x to ϕ(Wx), with W as a learnable weight matrix and ϕ as a nonlinear activation.
Optical feature extraction: Optical devices can outsource feature extraction by mapping data into compressed or expanded spaces that reveal features for machine-learning algorithms.This allows physical hardware to perform selected components without implementing a complete machine-learning model.
Optical feature extraction: Random light scattering generates high-dimensional random embeddings that approximately preserve distances, enabling dimensionality reduction and nearest-neighbor search.The same random matrix approximately preserves distances between vectors, supporting both data compression and efficient search.
Optical and quantum hardware: Quantum machine learning uses quantum states and interference to compute kernels, whose inner products can be processed by standard methods such as support vector machines.Quantum algorithms embed data in Hilbert space and measure overlaps between the resulting states.
Challenges and outlook: Current NISQ hardware often limits quantum-machine-learning experiments to proof-of-principle demonstrations, while some claimed exponential speedups rely on unfounded comparisons.Nevertheless, constant speedups may remain useful when computational gains matter.