Exterior complex scaling enables physics-informed neural networks for quantum scattering

The gist The exterior complex scaling enables physics-informed neural networks for nuclear reactions.

How it works

Exterior complex scaling (ECS) transforms scattering boundary conditions into exponentially decaying waves suitable for neural network solutions, enabling PINNs to solve nuclear reaction problems for the first time The method is validated on nucleon-nucleus scattering (n+40Ca at Elab = 20 MeV) with 21 partial waves, achieving phase shift accuracy of ∆δ ≲ 0.1◦ for the strongly absorbed channels (l ≤ 4) and ∆δ ≤ 0.60◦ for all channels up tol = 10 The source term is confined to the real axis, avoiding the need to analytically continue nuclear potentials into the complex plane

Theoretical Framework

The quantum mechanical description of nuclear scattering begins with the time-independent Schrödinger equation for the relative motion of two colliding nuclei The radial Schrödinger equation in the variable x (the physical radial coordinate) takes the form −ħ2/2µd2x/dx2 + ħ2/2l(l + 1)µx2 + VN(x) + VC(x) u(x) = E u(x) The wave function must satisfy the boundary condition u(0) = 0 to remain finite at the origin, and for scattering states at positive energy E > 0, it must match the appropriate asymptotic form at large distances

Exterior Complex Scaling Transformation

Exterior complex scaling provides an elegant solution to this boundary condition problem by analytically continuing spatial coordinates into the complex plane beyond a certain radius R0 When the radial coordinate r is rotated into the complex plane by an angle θ, outgoing waves that behave as eikr become exponentially damped as eikx(r) = eik[r cos θ+iy(r)] This transformation converts the scattering problem into one with squareintegrable (L2) boundary conditions: the wave function now decays exponentially at large distances and can be set to zero at a finite outer boundary The effect of this coordinate transformation on outgoing waves is crucial, as an outgoing wave eikx in the ECS region becomes eikx(r) = eik[r cos θ+iy(r)], where y(r) = Im[x(r)] increases approximately as (r − R0) sin θ for r > R0 + w

PINN Architecture and Loss Function

The PINN-ECS method combines exterior complex scaling, which transforms oscillatory scattering waves into decaying functions, with a physics-informed neural network that learns the scattered wave function by minimizing the differential equation residual The network architecture consists of a fully-connected feedforward network with r ∈ [0, Rmax] as input and two outputs representing the real and imaginary parts of the complex wave function, uR(r) and uI(r) The loss function combines the differential equation residual with an anchor term that prevents the trivial solution u sc = 0 The total loss is L = Lres + Lanchor, where Lres is the differential equation residual and Lanchor is a term designed to prevent the optimizer from finding the trivial solution u sc = 0

Validation and Results

The PINN-ECS method was validated on two benchmark systems: nucleon-nucleus scattering (n+40Ca at Elab = 20 MeV) and heavy-ion scattering (6Li+208Pb at 40 MeV) For n+40Ca, the mean ∆δ across all 21 spin-orbit channels is 0.09◦, and the wave function accuracy for low-l channels is below 0.15% For heavy-ion scattering, the mean ∆S across all 41 partial waves is 3.3 × 10−3, and the Rutherford ratio reproduces the COLOSS reference over three orders of magnitude The auto-adaptive anchor mechanism plays a role in achieving consistent accuracy across partial waves by engaging a linear warm-down for weak-source channels when sa < s∗a = 0.1

Conclusion and Future Directions

The PINN-ECS approach opens several directions for future research, building on the broader trend of integrating machine learning into nuclear physics The most immediate application is inverse problems: fitting optical potential parameters directly to experimental elastic scattering data by exploiting the end-to-end differentiability of the computational pipeline Extension to fewbody scattering problems where the meshfree nature of neural networks may help mitigate the curse of dimensionality faced by grid-based methods represents a longer-term goal The philosophy of transforming a difficult problem into one that matches the inductive biases of neural networks, namely smooth, bounded, decaying solutions, may prove fruitful in other areas of computational physics where boundary conditions pose fundamental challenges. The computational cost is dominated by the training iterations required for each partial wave, which requires on the order of a few minutes per channel.

1. The improved AI system can solve nuclear reaction problems by transforming oscillatory scattering boundary conditions into exponentially decaying waves that neural networks can represent naturally through Exterior Complex Scaling (ECS). This enables PINNs to solve nuclear reaction problems for the first time, as stated in Section I.

2. The system will be capable of solving inverse problems where end-to-end differentiability enables direct fitting of optical potential parameters, coupledchannel reactions, and few-body scattering where traditional grid methods face exponential scaling. This is achieved by exploiting the entire computational pipeline is end-to-end differentiable to perform gradient-based optimization of potential parameters to fit experimental data.

3. The AI can handle complex multi-output representations for coupled systems by leveraging the PINN's natural extension: PINNs naturally extend to multi-output representations, making coupled-channel calculations conceptually straightforward without explicitly constructing coupled matrix equations.

4. The system will be able to perform high-accuracy phase shift and S-matrix element extraction from complex scattering data. This is achieved by using multi-point S-matrix averaging and ensuring accuracy comparable to conventional solvers, such as achieving phase shift accuracy of ∆δ ≲ 0.1◦ for the strongly absorbed channels (l ≤ 4).

5. The system will incorporate an automatic mechanism to ensure unbiased results in challenging regimes. This is achieved by implementing an auto-adaptive anchor warm-down for weak-source channels, which prevents bias by driving the loss to zero when the converged solution minimizes the driven-equation residual without any anchor contribution.

6. The system will exhibit robustness against parameter tuning through a carefully designed loss function. This is achieved by using an anchor term that prevents the trivial solution u sc = 0 and setting the scale parameter automatically based on the source term: sa = ⟨VshortFl2⟩, which ensures that the anchor contribution remains comparable to the residual loss across all partial waves.

Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations, yet their application to nuclear scattering has been hindered by the oscillatory, non-decaying nature of scattering wave functions. In this work, I demonstrate that exterior complex scaling (ECS) transforms scattering boundary conditions into exponentially decaying waves suitable for neural network solutions, enabling PINNs to solve nuclear reaction problems for the first time. I develop a driven-equation formulation where the source term is confined to the real axis, avoiding the need to analytically continue nuclear potentials into the complex plane. The method is validated on nucleon-nucleus scattering (n+ 40 Ca at E lab=20 MeV) with 21 partial waves, achieving phase shift accuracy of Δδ 0.1 for the strongly absorbed channels (at most 4) and Δδ at most 0.60 for all channels up to = 10, when compared to conventional solvers. I further demonstrate the approach on heavy-ion scattering (6 Li+ 208 Pb at 40 MeV) with 41 partial waves and strong Coulomb effects, where an auto-adaptive anchor warm-down for weak-source channels yields a mean S-matrix accuracy of ΔS about 3 times 10-3 across the full angular momentum range, including the absorption-to-transparency transition region. This work establishes the foundation for extending PINNs to inverse problems where end-to-end differentiability enables direct fitting of optical potential parameters, coupled-channel reactions, and few-body scattering where traditional grid methods face exponential scaling.