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arXiv:2602.11193

Addressing the ground state of the deuteron by physics-informed neural networks

Lorenzo Brevi, Antonio Mandarino, Carlo Barbieri, Enrico Prati

physics.comp-phnucl-thquant-phSubmitted Jan 30, 2026• Announced Feb 2, 2026
arXivHTMLPDF

Abstract

Machine learning techniques have proven to be effective in addressing the structure of atomic nuclei. Physics$-$Informed Neural Networks (PINNs) are a promising machine learning technique suitable for solving integro-differential problems such as the many-body Schrödinger problem. So far, there has been no demonstration of extracting nuclear eigenstates using such method. Here, we tackle realistic nucleon-nucleon interaction in momentum space, including models with strong high-momentum correlations, and demonstrate highly accurate results for the deuteron. We further provide additional benchmarks in coordinate space. We introduce an expression for the variational energy that enters the loss function, which can be evaluated efficiently within the PINNs framework. Results are in excellent agreement with proven numerical methods, with a relative error between the value of the predicted binding energy by the PINN and the numerical benchmark of the order of $10^{-6}$. Our approach paves the way for the exploitation of PINNs to solve more complex atomic nuclei.

DOI
10.1016/j.physletb.2026.141050

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Machine learning techniques have proven to be effective in addressing the structure of atomic nuclei. Physics−-−Informed Neural Networks (PINNs) are a promising machine learning technique suitable for solving integro-differential problems such as the many-body Schrödinger problem. So far, there has been no demonstration of extracting nuclear eigenstates using such method. Here, we tackle realistic nucleon-nucleon interaction in momentum space, including models with strong high-momentum correlations, and demonstrate highly accurate results for the deuteron. We further provide additional benchmarks in coordinate space. We introduce an expression for the variational energy that enters the loss function, which can be evaluated efficiently within the PINNs framework. Results are in excellent agreement with proven numerical methods, with a relative error between the value of the predicted binding energy by the PINN and the numerical benchmark of the order of 10−610^{-6}10−6. Our approach paves the way for the exploitation of PINNs to solve more complex atomic nuclei.
Quantum many-body systems are of great interest for many research areas, including physics, biology, and chemistry. However, their simulation is extremely challenging, due to the exponential growth of the Hilbert space with system size, making it exceedingly difficult to parameterize the wave functions of large systems by using exact methods. Neural networks and machine learning, in general, are a way to face this challenge. For instance, methods like tensor networks and neural quantum states are being investigated as promising tools to obtain the wave function of a quantum mechanical system. In this tutorial, we focus on a particularly promising class of deep learning algorithms. We explain how to construct a Physics-Informed Neural Network (PINN) able to solve the Schrödinger equation for a given potential, by finding its eigenvalues and eigenfunctions. This technique is unsupervised, and utilizes a novel computational method in a manner that is barely explored. PINNs are a deep learning method that exploit automatic differentiation to solve integro-differential equations in a mesh-free way. We show how to find both the ground and the excited states. The method discovers the states progressively by starting from the ground state. We explain how to introduce inductive biases in the loss to exploit further knowledge of the physical system. Such additional constraints allow for a faster and more accurate convergence. This technique can then be enhanced by a smart choice of collocation points in order to take advantage of the mesh-free nature of the PINN. The methods are made explicit by applying them to the infinite potential well and the particle in a ring, a challenging problem to be learned by an artificial intelligence agent due to the presence of complex-valued eigenfunctions and degenerate states
Predicting the structure of quantum many-body systems from the first principles of quantum mechanics is a common challenge in physics, chemistry, and material science. Deep machine learning has proven to be a powerful tool for solving condensed matter and chemistry problems, while for atomic nuclei it is still quite challenging because of the complicated nucleon-nucleon interactions, which strongly couple the spatial, spin, and isospin degrees of freedom. By combining essential physics of the nuclear wave functions and the strong expressive power of artificial neural networks, we develop FeynmanNet, a deep-learning variational quantum Monte Carlo approach for ab initio nuclear structure. We show that FeynmanNet can provide very accurate solutions of ground-state energies and wave functions for 4He,6Li^{4}\mathrm{He}, ^{6}\mathrm{Li}4He,6Li, and even up to 16O^{16}\mathrm{O}16O as emerging from the leading-order and next-to-leading-order Hamiltonians of pionless effective field theory. Compared to the conventional diffusion Monte Carlo approaches, which suffer from the severe inherent fermion-sign problem, FeynmanNet reaches such a high accuracy in a variational way and scales polynomially with the number of nucleons. Therefore, it paves the way to a highly accurate and efficient ab initio method for predicting nuclear properties based on the realistic interactions between nucleons.
To distill the essential elements of nuclear binding, we seek the simplest Hamiltonian capable of modeling atomic nuclei with percent-level accuracy. A critical aspect of this endeavor consists of accurately solving the quantum many-body problem without incurring an exponential computing cost with the number of nucleons. We address this challenge by leveraging a variational Monte Carlo method based on a highly expressive neural-network quantum state ansatz. In addition to computing binding energies and charge radii of nuclei with up to A=20 nucleons, by evaluating their magnetic moments, we demonstrate that neural-network quantum states are able to correctly capture the self-emerging nuclear shell structure. To this end, we introduce a novel computational protocol based on adding an external magnetic field to the nuclear Hamiltonian, which allows the neural network to learn the preferred polarization of the nucleus within the given magnetic field.
Physics-Informed Neural Networks (PINNs) offer an effective framework for solving partial differential equations by incorporating governing physical laws into the learning process. When applied to quantum mechanical differential equations such as the Schr¨odinger equation, this approach produces smooth wavefunctions suitable for computing both conventional observables and Bohmian quantities. In this work, a PINNs based framework is developed to solve the one-dimensional time-dependent Schr¨odinger equation for the Quantum Har monic Oscillator, a fundamental model with broad relevance in quantum and plasma physics. A fully connected neural network is used to represent the real and imaginary parts of the wavefunction, with the Schr¨odinger equation enforced through the loss function together with appropriate initial and boundary conditions. Automatic differentiation is employed to compute the required derivatives and to extract physical quantities from the learned solution. The method accurately reproduces the ground-state wavefunction and yields a ground-state energy in agreement with the analytical result. The Bohm quantum potential obtained from the network also matches the corresponding analytical expression, demon strating that the essential spatial structure of the quantum amplitude is well captured. These results demonstrate that PINNs constitute a reliable and conceptually transparent framework for solving the time-dependent Schr¨odinger equation. Within this differential equation based setting, PINNs also enable systematic exploration of Bohmian aspects of quantum systems, where Bohmian quantities such as the quantum potential provide physical insight into confinement effects and the emergence of effective potentials.
Quantum Fisher Information (QFI) sets the ultimate precision limit for parameter estimation and is therefore a central quantity in quantum metrology. In time-dependent many-body systems, however, maximizing QFI is a highly non-trivial task due to the combined effects of non-commutativity, control complexity, and the exponential growth of the Hilbert space. In this work, we present a physics-informed neural network (PINN) framework to address this problem through the learning of counter-diabatic quantum dynamics. Our approach combines a variational PINN formulation with a Magnus-expansion treatment of time-ordered evolution, enabling the adiabatic gauge potential and the scheduling function to be inferred directly from the underlying physics while enforcing the Euler-Lagrange structure of the protocol. The method is applied to several families of driven spin Hamiltonians, including nearest-neighbor, dipolar, and trapped-ion-inspired interactions, for systems of up to six qubits. The numerical results show that the proposed framework systematically improves over reference solutions based only on the Euler-Lagrange condition, yielding high normalized QFI together with favorable fidelity and extremal-balance metrics while preserving small phsical residuals. The analysis further shows that learning the scheduling function provides a clear performance advantage in most cases, and reveals non-trivial finite-size effects, with q=3q=3q=3 emerging as a particularly challenging regime. Although scalability remains limited by the exponential growth of the operator space and by automatic-differentiation costs, the results demonstrate that PINNs constitute a viable and physically grounded route for learning metrologically optimal control strategies in interacting quantum systems.
Physics-informed neural networks (PINNs) have emerged as a versatile approach for solving nonlinear partial differential equations (PDEs), yet achieving high accuracy efficiently using these techniques remains challenging for high-dimensional or multiscale systems. Here, we present a hybrid quantum-classical framework that enhances Quantum PINNs (QPINNs) through adaptive collocation point sampling and loss-aware attention mechanisms. By dynamically prioritizing points in regions with large PDE residuals or steep solution gradients, our method mitigates the spectral bias inherent in conventional PINNs. Current Quantum Physics-Informed Neural Networks are commonly assumed to be limited by the expressive power of quantum circuits. In our work, we observed that, across diverse differential equations, optimization - not only expressivity - can be an important bottleneck. Furthermore, a trainable loss-weighting scheme balances contributions from physics residuals, boundary conditions, and data fidelity during training. Integrating these strategies with quantum computing techniques (including variational quantum circuits and quantum gradient estimation) can yield at least a 60% improvement in solution accuracy under specific regimes for benchmark fluid flows and reaction-diffusion systems. Finally, we argue that merely increasing model expressivity is insufficient for resolving complex PDEs via QPINNs, as they remain constrained by the structural optimization limitations of classical PINNs. This framework provides a scalable pathway for quantum-enhanced scientific machine learning, bridging physics-based modeling with emerging quantum computational capabilities.