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

This work demonstrates that exterior complex scaling transforms non-decaying scattering wave functions into exponentially decaying forms, enabling physics-informed neural networks to accurately solve nuclear scattering problems for the first time and paving the way for efficient inverse problems and complex reaction modeling.

The Gist

The Big Problem: The "Endless Echo"

Imagine you are trying to predict how a ball bounces off a wall. In the world of atoms and nuclei, this "ball" is a particle, and the "wall" is another nucleus. When they collide, they don't just stop; they scatter.

In physics, the math describing this scattering is like a wave that never stops moving. It goes on forever, oscillating back and forth like a sound wave in an endless canyon.

For decades, scientists have used standard computer programs to solve these equations. These programs work like a grid or a ladder: they step from one point to the next. But because the wave never stops, the computer has to keep stepping forever to get the answer right. If you stop the ladder too early, you get a wrong answer (like an echo bouncing off a wall that shouldn't be there).

Recently, a new type of computer program called a Physics-Informed Neural Network (PINN) became popular. Think of a PINN as a super-smart student who learns by looking at the rules of the game (the physics equations) rather than stepping through a grid. PINNs are great at solving problems where things settle down and stop (like heat cooling down). But they fail miserably at nuclear scattering because the "wave" never settles down; it just keeps oscillating forever. The student gets confused and can't find the answer.

The Solution: The "Complex Mirror"

The author of this paper, Jin Lei, found a clever trick to make the neural network student understand the problem. He used a mathematical technique called Exterior Complex Scaling (ECS).

Imagine the nuclear collision happens in a room.

1. The Real Room: Inside the room (close to the nucleus), the physics is normal. The particle bounces around, and the walls are real.
2. The Complex Mirror: Once the particle leaves the room and enters the "outside," the author turns the floor into a mirror that tilts the world into a different dimension (the complex plane).

In this tilted, "complex" world, the endless oscillating wave suddenly transforms. Instead of bouncing back and forth forever, it starts to fade away like a sound dying out in a thick fog. It becomes an "exponentially decaying wave."

Now, the neural network student is happy! It sees a wave that fades away and stops. It can easily learn the rules because the problem looks like the "settling down" problems it is good at solving.

The "Driven" Trick: Separating the Noise

To make this work perfectly, the author also changed how the problem is asked.

Instead of asking the neural network to figure out the entire wave from scratch, he split it into two parts:

1. The Known Part: A "background" wave that the computer already knows how to calculate (like a standard sound wave).
2. The "Driven" Part: The messy, interesting part caused by the collision.

The author set up the math so that the "messy" part only exists where the nuclei actually touch (the real room). Once the particle leaves that room, the "messy" part is forced to be zero. This means the neural network only has to learn the messy part in the real room and then watch it fade away in the complex mirror. It doesn't have to guess what happens in the infinite distance; the math forces it to fade out.

The Results: Testing the New Method

The author tested this new method on two different scenarios to prove it works:

1. The Light Test (Neutron + Calcium): He simulated a neutron hitting a Calcium nucleus. The results were incredibly accurate, matching the best traditional computer methods almost perfectly. The difference was so small it was barely noticeable (less than one-tenth of a degree in the angle of the bounce).
2. The Heavy Test (Lithium + Lead): He simulated a heavier collision between Lithium and Lead. This is harder because the electric repulsion between them is huge. The method still worked, accurately predicting how the particles scattered, even in the tricky "gray area" where the particles barely touch.

Why This Matters (According to the Paper)

The paper claims this is a breakthrough because:

• It works where others failed: It's the first time neural networks have successfully solved these specific nuclear scattering problems.
• It's "End-to-End": Because the whole process is built on a neural network, you can tweak the input (like the strength of the nuclear force) and the computer instantly knows how the output changes. This is great for "inverse problems"—figuring out what the nucleus looks like based on how particles bounce off it.
• It handles the "Hard" stuff: It can deal with complex shapes and multiple particles without needing to build a rigid grid, which usually causes computers to crash when things get too complicated.

In short: The author built a mathematical "funnel" (the complex scaling) that turns an impossible, endless wave problem into a simple, fading wave problem that a modern AI can solve easily and accurately.

Technical Summary

Technical Summary: Exterior Complex Scaling Enables Physics-Informed Neural Networks for Nuclear Reactions

Problem Statement
Physics-informed neural networks (PINNs) have emerged as a powerful tool for solving differential equations across various domains. However, their application to nuclear scattering has been fundamentally hindered by the nature of scattering wave functions. Unlike bound-state problems where wave functions decay exponentially, scattering wave functions remain oscillatory at all radii, satisfying radiation boundary conditions (Sommerfeld conditions) at infinity. Traditional PINNs operate on finite computational domains and require boundary conditions that can be specified independently of the solution. The oscillatory, non-decaying nature of scattering waves prevents the imposition of simple Dirichlet or Neumann conditions at finite boundaries; truncating the domain introduces spurious reflections that contaminate the solution. This incompatibility between oscillatory radiation conditions and finite-domain computation has previously blocked the application of PINNs to nuclear reaction problems.

Methodology
This work introduces a framework combining Exterior Complex Scaling (ECS) with PINNs to overcome the boundary condition obstacle. The methodology consists of three primary components:

1. Exterior Complex Scaling (ECS): The radial coordinate $r$ is analytically continued into the complex plane beyond a scaling radius $R_0$. For $r > R_0$, the coordinate is rotated by an angle $\theta$. This transformation converts outgoing oscillatory waves ($e^{ikr}$) into exponentially damped functions ($e^{ikr \cos \theta} e^{-kr \sin \theta}$). Consequently, the scattering problem is transformed into one with square-integrable ($L^2$) boundary conditions, allowing the wave function to be set to zero at a finite outer boundary without spurious reflections. Crucially, the ECS transformation keeps the nuclear interaction region on the real axis, avoiding the need to analytically continue the nuclear potential into the complex plane.

2. Driven-Equation Formulation: To further simplify the learning task, the total wave function $u(r)$ is decomposed into a known incident Coulomb wave $u_{in}(r)$ and a scattered wave $u_{sc}(r)$. The scattered wave satisfies an inhomogeneous driven equation where the source term is confined entirely to the real axis (within the nuclear range). In the ECS region ($r > R_0$), the source term vanishes, and the scattered wave satisfies a homogeneous equation with exponentially decaying boundary conditions. This formulation allows the neural network to learn a function that decays to zero, rather than an oscillatory function.

3. Neural Network Architecture and Training:

   • Architecture: A fully connected feedforward network with sinusoidal activation functions ($\sin(z)$) is used to represent the real and imaginary parts of the scattered wave.
   • Boundary Conditions: Hard-coded multiplicative factors enforce $u_{sc}(0)=0$ and $u_{sc}(R_{max})=0$. A sigmoid-capped boundary condition factor is introduced to prevent numerical overflow for high angular momentum ($\ell$) channels, where the standard $r^{\ell+1}$ behavior would require the network to output vanishingly small values.
   • Loss Function: The loss combines the residual of the driven equation with an anchor term to prevent the trivial solution ($u_{sc}=0$).
   • Auto-Adaptive Anchor Warm-Down: For channels with weak source terms (high $\ell$ or strong Coulomb suppression), a fixed anchor can bias the solution. The authors implement an auto-adaptive mechanism where the anchor weight is linearly "warmed down" to zero for channels where the source magnitude falls below a threshold. This ensures that the final solution is driven solely by the residual loss, eliminating artificial absorption biases in near-transparent channels.
   • Training Strategy: A two-stage training process (Adam optimizer followed by L-BFGS) is employed. For heavy-ion systems, a causal training strategy is used, progressively expanding the training region from the nuclear interior outward to ensure stable convergence.

Key Contributions

• First Application of PINNs to Nuclear Reactions: This work demonstrates the first successful application of PINNs to nuclear scattering problems by resolving the boundary condition incompatibility via ECS.
• Driven-Equation Formulation: The development of a source-confined driven equation that avoids analytic continuation of nuclear potentials.
• Technical Innovations: Introduction of the sigmoid-capped boundary factor for high-$\ell$ stability and the auto-adaptive anchor warm-down for unbiased treatment of weak-source channels.
• End-to-End Differentiability: The framework establishes a pipeline where the entire calculation is differentiable, enabling direct gradient-based optimization of optical potential parameters (inverse problems).

Results
The method was validated on two benchmark systems against the established COLOSS code (which uses Lagrange-mesh discretization with complex scaling):

1. Neutron-Nucleus Scattering ($n + ^{40}\text{Ca}$ at 20 MeV):

   • Calculated 21 partial waves (including spin-orbit partners up to $\ell=10$) using the Koning-Delaroche global optical potential.
   • Accuracy: Phase shift errors ($\Delta\delta$) were $\lesssim 0.1^\circ$ for strongly absorbed channels ($\ell \le 4$) and $\le 0.60^\circ$ for all channels up to $\ell=10$. The mean error across all channels was $0.09^\circ$.
   • Wave Function: The relative RMS difference between PINN and reference solutions was below $0.15\%$ for low-$\ell$ channels and below $2\%$ even for the worst-case channel.

2. Heavy-Ion Scattering ($^6\text{Li} + ^{208}\text{Pb}$ at 40 MeV):

   • Calculated 41 partial waves with strong Coulomb effects ($\eta \approx 15$).
   • Accuracy: The mean S-matrix accuracy across all partial waves was $|\Delta|S_\ell|| \approx 3 \times 10^{-3}$. In the challenging absorption-to-transparency transition region ($\ell \approx 25\text{--}30$), the error remained below $0.007$.
   • Observables: The calculated Rutherford ratio reproduced the reference solution over three orders of magnitude, capturing the characteristic Coulomb-nuclear interference pattern.

Significance and Claims
The paper claims that this work establishes the foundation for extending PINNs to nuclear reaction physics. The primary significance lies in enabling end-to-end differentiability, which allows for the direct fitting of optical potential parameters to experimental data, a task that traditionally requires expensive outer-loop optimizations. Furthermore, the mesh-free nature of the method offers a potential pathway to solving coupled-channel reactions and few-body scattering problems (e.g., Faddeev equations) where traditional grid-based methods face exponential scaling costs.

The authors note that the method is currently limited to reactions where physics is confined to radii smaller than the ECS scaling radius (elastic, inelastic, and transfer reactions). Processes dominated by long-range electromagnetic transitions (e.g., sub-barrier Coulomb excitation) cannot be treated directly within this framework. The computational cost is currently higher than conventional integration methods (minutes per channel vs. seconds), but the authors argue this is acceptable for research applications requiring differentiability or mesh-free formulations.