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

arXiv:2602.11193 · physics.comp-ph, nucl-th, quant-ph · Submitted 2026-01-30 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Addressing the ground state of the deuteron by physics-informed neural networks".

Mira: The gist The PINN framework allows for addressing realistic nucleon-nucleon interactions in momentum space to obtain highly accurate results for the deuteron ground state,

Kai: First, who's behind it and why it matters.

Paper summary: Kai: So we've covered how this paper, "Addressing the ground state of the deuteron by physics-informed neural networks," uses PINNs to tackle realistic nucleon-nucleon interactions in momentum space to get highly accurate results for the deuteron ground state.

Mira: The authors are essentially showing that they can solve a manybody Schrödinger problem using this framework, introducing a new strategy for computing eigenenergies that extends the standard PINN approach. They provide examples of analogous simulations in coordinate space as well.

Lev: From what I see, the main thing this paper does is prove that PINNs can be applied to an ab initio nuclear structure problem, which is a big step because it's not just solving some simple toy model.

Kai: The authors conclude that their approach paves the way for exploiting PINNs to solve more complex atomic nuclei in the future, even though they only showed the deuteron here.

Mira: It suggests that this method of integrating all possible physical knowledge into a loss function is a viable path forward when dealing with complex quantum systems where you might lack target data.

Lev: The limitation they state is that their implementation doesn't extend to full three dimensional space yet, so we know exactly what the next engineering hurdle looks like for taking this from the deuteron to something more substantial.

Conclusion: Kai: So, we've seen how they took realistic nucleon-nucleon interactions in momentum space and used physics to train a neural network to get highly accurate results for the deuteron ground state.

Mira: It’s a specific problem, solving that manybody Schrödinger equation using these PINNs, which is really pushing what you can do with these architectures.

Lev: From my side, I'm thinking about how much overhead that training process adds when we try to run this kind of calculation on actual quantum hardware.

Kai: Yeah, it’s a demonstration of feasibility, showing the method works for this simple system before we try something more complex.

Mira: It means these physics-informed networks aren't just theoretical exercises; they have the potential for ab initio nuclear structure problems.

Lev: But you gotta remember, this was done in momentum space simulations with specific interactions like N4LO and CD-Bonn, not necessarily the full three dimensions we want to aim for yet.

Kai: That’s right; they showed excellent agreement with numerical benchmarks, with a relative error down to about ten-six.

Mira: That level of precision is what makes this approach so compelling for solving these complex quantum systems.

Lev: And it sets a benchmark for how we can use machine learning to find those bound states in nuclear physics.

Kai: It opens the door, then, to using PINNs as a tool instead of just a black box when dealing with the manybody problem.

Mira: Exactly; integrating the variational principle directly into the loss function is that crucial physical constraint they're exploiting.

Lev: It shows that we can start leveraging these tools for nuclear structure right now, even if it’s just for testing concepts like this deuteron example.

Dipartimento di Fisica "Aldo Pontremoli" UniversitaÓ di Milano International Centre for Theory of Quantum Technologies University of Gdansk INFN Seziona di Milano

physics.comp-ph, nucl-th, quant-ph

Submitted: 2026-01-30

Updated: 2026-01-30

DOI: 10.1016/j.physletb.2026.141050

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 72/100

The gist: The gist The PINN framework allows for addressing realistic nucleon-nucleon interactions in momentum space to obtain highly accurate results for the deuteron ground state, demonstrating feasibility

Key concepts

Physics-Informed Neural Networks (PINNs): PINNs are a machine learning technique that solves differential equations by incorporating the governing physical laws directly into the network's loss function. This allows the neural network to learn solutions to complex problems, like the Schrödinger equation, without needing extensive training data or explicit prior knowledge of all physical constraints.
Loss Function Components (LPINN): The overall loss function for PINNs is a combination of several terms: LPDE (for solving the differential equation), Ldata (for supervised learning), LBCs (for boundary conditions in coordinate space), Lnorm (to enforce normalization), and Lvar (to implement the variational principle). This comprehensive loss function ensures the network learns a physically consistent wavefunction.
Momentum Space Simulations: The study tested two realistic nucleon-nucleon interaction models in momentum space: N4LO, which uses a cutoff of 550 MeV/c, and CD-Bonn, which extends up to several GeV/c. These simulations showed that the accuracy of the results is largely dictated by the potential model itself rather than the PINN method.
Variational Energy Loss (Lvar): The variational principle is implemented in the loss function through Lvar. This term forces the network's predicted wavefunction to adhere to physical constraints derived from minimizing energy, ensuring that the resulting solution behaves qualitatively like a bound L=0 wave, concentrating probability where the nuclear interaction is most attractive.

Terminology

Benchmark — 2× in this paper · explained in 2 episodes
Loss Function — 1× in this paper · explained in 2 episodes

Summary

The gist The PINN framework allows for addressing realistic nucleon-nucleon interactions in momentum space to obtain highly accurate results for the deuteron ground state, demonstrating feasibility for ab initio nuclear structure problems.

Physics-Informed Neural Network Framework

The PINN framework is introduced to solve integro-differential problems such as the manybody Schrödinger problem by integrating all possible physical knowledge into the loss function of the network. The general form for its loss function is LPINN = LPDE + Lphys + Ldata, which encodes all prior knowledge about the physical system. The PDE term LPDE is defined as a mean squared error or sum of squared errors of the differential equation.

Loss Function Components

The overall PINN loss function is summarized as LPINN = αS ch LS chrod + αint Lint + αBCs LBCs+ αnorm Lnorm + αvar Lvar, where appropriate weights are added to each term.

The components of the loss function include:

LPDE, which is the mean squared error or sum of squared errors of the differential equation.
Ldata, which is the standard training loss of a supervised problem.
LBCs, which encodes physical constraints due to boundary conditions in coordinate space.
Lnorm, which enforces the normalization condition Lnorm = Nc I[ψ] − ln (I[ψ]) − 1 + ν(0) (8).
Lvar, which implements the variational principle. The network was trained using the overall loss from Eq. (12) computed over 4097 collocation points and with the weights from Table 1. The converged wavefunction follows the qualitative behavior of a bound L=0 wave, with an enhanced probability where the interaction is most attractive.

Momentum Space Simulations

In momentum space simulations, two realistic models of the nuclear interaction were tested:

The N4LO interaction, which uses a cutoff on the relative momentum of 550 MeV/c.

For the N4LO interaction, a uniform linear mesh was used in the interval [0, kmax] with kmax ≈ 1400 MeV/c.

For the CD-Bonn interaction, a mesh extending up to several GeV/c was used with kmax ≈ 8573 MeV/c.

Comparison and Conclusion

The comparison between interactions shows that the error is maximum at 0.01 for the very simple Minnesota potential, and minimum for the more modern N4LO, at −5.96 × 10−6. The main source of error is from the potential itself and not the PINN method. The paper concludes that the PINN framework demonstrates feasibility for microscopic nuclear simulation, paving the way for a completely new way to utilize novel machine learning tools for theoretical physics. Future work should focus on extending this implementation to full three dimensional space.

How it works

The approach leverages the property of exceptionally efficient universal approximators in neural networks to represent wavefunctions of complex quantum systems. This NQS approach can be referred to as science-driven learning because no training or target data is provided to learn about many-body correlations.

The PINN framework addresses the limitation of relying on only one piece of physics information, the variational principle, by integrating all possible physical knowledge into the loss function. This allows for training neural networks with noisy or very little data and even in the absence of labeled data. The differential equation loss LS chrod imposes that the solution solves the Schrödinger’s equation H ψ⟩ = E ψ⟩ <ref:2602.11193n, The PDE term of the loss is then, LS chrod = X Nc i=0 Hψ − Eψ(xi).

Improvements for AI systems

The improved AI system can perform high-accuracy ab initio nuclear structure calculations by solving integro-differential problems like the manybody Schrödinger problem using Physics-Informed Neural Networks (PINNs). This allows for extracting highly accurate results for systems like the deuteron, 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.
The system can handle realistic nucleon-nucleon interactions in momentum space, including those with strong high-momentum correlations, which is critical for accurately modeling complex nuclear forces like N4LO or CD-Bonn models.
The AI can be trained to solve the Schrödinger equation by minimizing a comprehensive loss function: LPINN = αS ch LS chrod + αint Lint + αBCs LBCs+ αnorm Lnorm + αvar Lvar. This integrated approach ensures physical constraints (like boundary conditions and normalization) are enforced alongside solving the differential equation.
The system can utilize a novel strategy for eigenvalue computation, incorporating the variational principle via Lvar = min 0, (bE)c + d(t)e(a(E−E0)), which is designed to lead the network toward the lowest energy states.
The resulting AI model can provide insights into physical behavior during training by monitoring partial losses; for example, observing that the large value of LBCs in this region indicates that the bound state boundary conditions are not met and scattering states are still admixed into the wavefunction.