This paper introduces a neural-network method that can produce accurate nuclear wavefunctions as the parameters of the nuclear force are varied. The authors train a single “parametric neural quantum state” (NQS) that takes the interaction parameters as inputs. Once trained, that one neural network can give the quantum wavefunction for any set of those parameters in its training range. From the wavefunction, many static observables can be computed quickly without retraining the network for each new interaction.

The team tested the idea on two simple nuclear problems: the deuteron ground state (a bound state of a proton and neutron) and neutron–proton elastic scattering. They used local nuclear interactions derived from chiral effective field theory (EFT) up to next-to-next-to-leading order (N^2LO, commonly called third order). In their setup they varied the short-range interaction strengths, known as low-energy couplings (LECs), while keeping the longer-range pion-exchange pieces fixed. The neural network takes as input the particle positions, the LECs, and — for scattering — the incoming momentum, and it outputs a complex-valued wavefunction that includes spin information.

At a high level, the method works because neural networks can compactly represent complicated wavefunctions. For a bound state the network is trained by minimizing the expected energy. For scattering states it is trained by minimizing how much the network’s output violates the time-independent Schrödinger equation. Because the LECs are part of the network inputs, the trained model behaves like a fast surrogate solver: you can query the network for many different interaction parameter sets and then compute observables from the returned wavefunction.

To check accuracy the authors built neural scattering states (NSSs) and compared phase shifts — a standard scattering observable — against results from a direct solution of the Lippmann–Schwinger equation. They evaluated 10,000 samples of LEC sets drawn from a published posterior distribution and showed the distribution of relative errors between the neural method and the direct solver. The comparison indicates generally high fidelity, though a fraction of samples show relative errors larger than 2 percent. The paper provides these comparisons to quantify the method’s performance.