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

Jin Lei

arXiv:2602.04553v2nucl-thphysics.comp-ph

TL;DR

Oscillatory scattering boundary conditions have hindered PINNs from solving nuclear scattering problems on finite domains. This work combines exterior complex scaling with a driven-equation formulation that keeps the source on the real axis. It reports accurate results on nucleon–nucleus and heavy-ion benchmarks, with a scope boundary for long-range electromagnetic transitions.

  • Problem

    PINNs have struggled with scattering because oscillatory Sommerfeld conditions depend on the unknown S-matrix and cannot be imposed at a finite boundary without spurious reflections.

    Source: §II · p. 3

  • Method

    The method combines exterior complex scaling, which makes outgoing waves decay, with a driven equation whose source is confined to the real axis.

    Source: §II · p. 4

  • Results

    The benchmarks report nucleon–nucleus phase-shift errors of Δδ ≤ 0.60° across channels and a mean heavy-ion S-matrix accuracy of 3.3 × 10^-3 across 41 partial waves.

    Source: §IV · p. 8, §VI · p. 12

  • Takeaways & Limitations

    The validated approach establishes a foundation for future inverse-potential fitting, coupled-channel calculations, and few-body scattering studies.

    Source: §VI · pp. 12–13

    The method is limited to reactions whose relevant physics is confined inside R0 and does not directly treat long-range electromagnetic transitions dominated by large impact parameters.

    Source: §V · p. 12

Abstract

from arXiv · show

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 $E_{\text{lab}}=20$~MeV) with 21 partial waves, achieving phase shift accuracy of $\Delta\delta \lesssim 0.1^\circ$ for the strongly absorbed channels ($\ell \leq 4$) and $\Delta\delta \leq 0.60^\circ$ for all channels up to $\ell = 10$, 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 $|\Delta|S_\ell|| \approx 3 \times 10^{-3}$ 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.

I. INTRODUCTION

Nuclear scattering is difficult for finite-domain PINNs because oscillatory radiation boundary conditions depend on the unknown scattering solution. The paper combines exterior complex scaling with a driven-equation formulation and validates the approach on nucleon–nucleus and heavy-ion benchmarks.

I. INTRODUCTION

  • Oscillatory Sommerfeld conditions depend on the unknown phase shift and cannot be imposed at a finite boundary without spurious reflections, obstructing PINN scattering solutions.For long-range Coulomb interactions, matching is also delicate because the Coulomb field modifies asymptotic behavior.Source: §I · pp. 1–2
  • Exterior complex scaling turns outgoing waves into exponentially damped waves, enabling finite-boundary conditions while preserving the interaction region on the real axis.The transformation acts beyond the potential range, where the outgoing-wave amplitude decays exponentially.Source: §I · p. 2
  • The approach is motivated by differentiable potential fitting, coupled-channel representations, and possible few-body applications, rather than replacing efficient conventional single-channel solvers.Grid-based methods face exponential computational cost for few-body scattering.Source: §I · p. 2
  • The driven-equation formulation confines the source to the real axis and adds high-ℓ boundary conditioning, adaptive warm-down, and multi-point S-matrix averaging.The warm-down targets weak-source high-ℓ channels, where the centrifugal barrier suppresses the source term.Source: §I · p. 2
  • The benchmarks cover 21 partial waves for n+40Ca at 20 MeV and 41 partial waves for 6Li+208Pb at 40 MeV, including strong Coulomb effects.The nucleon case is compared with COLOSS; the heavy-ion case reports the Rutherford ratio and Coulomb–nuclear interference pattern.Source: §I · p. 2

II. THEORETICAL FRAMEWORK

The theoretical framework combines the radial partial-wave scattering equation with exterior complex scaling and a driven equation. This makes the outer solution decay while keeping the short-range source on the real axis, after which the S-matrix is extracted by matching on the real axis.

II. THEORETICAL FRAMEWORK

  • Each partial wave obeys a radial Schrödinger equation with a short-range complex optical potential, Coulomb interaction, and angular-momentum barrier.The wave function vanishes at the origin; the potential’s imaginary part accounts for flux loss to non-elastic channels.Source: §II · pp. 2–3
  • Conventional solvers integrate outward and match to asymptotic Coulomb forms, whereas finite-domain PINNs cannot directly impose the unknown incoming–outgoing radiation condition.The standard matching radius is typically about 15–20 fm for nucleon–nucleus scattering.Source: §II · p. 3
  • ECS smoothly maps the real parameter coordinate to a complex physical coordinate beyond the scaling radius, with a transition region of width w.The mapping and its first derivative are continuous, and the Jacobian approaches e^iθ beyond the transition region.Source: §II · p. 3
  • The transformed outgoing wave decays exponentially, and the outer-boundary damping factor is required to fall below approximately 10^-3.The decay depends on the wave number, the distance from R0, and sin θ.Source: §II · p. 3
  • The driven equation separates incident and scattered waves, confines its source to the real axis, and leaves a homogeneous equation with decaying boundary conditions in the ECS region.The nuclear potential is set to zero beyond the scaling radius, so it need not be analytically continued into the complex plane.Source: §II · p. 4
  • The S-matrix is extracted by Wronskian matching at a real-axis radius below R0 and averaged over five radii to reduce sensitivity to local numerical fluctuations.The direct S-matrix representation remains well-defined for absorptive optical potentials.Source: §II · p. 4

III. COMPUTATIONAL METHOD

PINN-ECS combines exterior complex scaling with a physics-informed neural network to learn the scattered wave from the differential-equation residual. Its implementation hard-codes boundary conditions and uses a sigmoid-capped factor, adaptive anchoring, and staged training to address high-angular-momentum and weak-source channels.

III. COMPUTATIONAL METHOD

  • PINN-ECS combines exterior complex scaling with a PINN for the scattered wave, using a four-layer sinusoidal network and hard-coded boundary conditions.The network takes the radial coordinate as input and multiplies its raw output by a boundary-condition factor.Source: §III · pp. 4–5, Figure 1 · p. 6
  • A sigmoid-capped boundary factor preserves r^(ℓ+1) behavior near the origin while preventing high-ℓ growth from compressing the network’s interior dynamic range.For ℓ>0, the factor saturates beyond Rnuc; the ℓ=0 factor is not capped.Source: §III · p. 5
  • The loss adds an anchor penalty to the differential-equation residual to prevent the trivial zero scattered wave, with its scale set from source magnitude.The anchor penalty vanishes when the scattered-wave magnitude is large relative to its scale.Source: §III · p. 5
  • Weak-source channels automatically warm down the anchor to zero, leaving the final Adam epochs and all L-BFGS optimization driven by the residual alone.The schedule is used below a source-scale threshold, including heavy-ion channels in the absorption-to-transparency region.Source: §III · p. 5
  • At ℓ=25 for 6Li+208Pb, anchor-parameter scans produced spreads of 3×10^-5 in |S25| and 8×10^-4° in δ25.For the n+40Ca j15/2, ℓ=7 channel, the corresponding spreads were 4×10^-4 in |S| and 0.10° in δ.Source: §III · p. 6
  • Collocation points are concentrated across the nuclear interior, intermediate region, and ECS region; reducing their budgets below about 1500 for n+40Ca or 3000 for 6Li+208Pb degrades high-ℓ phase shifts.The stated allocations are 50%, 20%, and 30%, respectively.Source: §III · p. 6
  • Training uses Adam followed by L-BFGS, with larger collocation and epoch budgets for 6Li+208Pb than for n+40Ca.The method also increases optimization budgets for initial high-ℓ channels and transfers converged weights between spin-orbit partners when applicable.Source: §III · p. 7
  • For heavy-ion systems, causal training expands the fitted radial region from the nuclear interior outward so the interior solution is established before the asymptotic region.Points beyond the expanding boundary contribute less to the loss during the warmup.Source: §III · p. 7

IV. RESULTS

The results compare PINN-ECS with COLOSS for neutron–calcium and lithium–lead scattering, including phase shifts, S-matrix magnitudes, and wave functions. Accuracy is reported across partial waves, with auto-adaptive anchor warm-down used for weak-source channels.

n+40Ca benchmark

  • For n+40Ca, S-matrix magnitudes agree with COLOSS to within ∆|S|≤0.005 across all 20 spin-orbit channels, including high-ℓ channels with suppressed source terms.Table I includes representative channels from interior, surface, and high-angular-momentum regions.Source: Table I · p. 8, §IV · p. 8
  • 0.20% is the mean relative RMS wave-function difference across 21 n+40Ca channels versus Runge-Kutta; the high-ℓ tail remains at the 0.1% level.The comparison evaluates interior, surface, and outer radial regions.Source: Table II · p. 8, §IV · p. 8

Benchmark systems

  • The 6Li+208Pb benchmark covers ℓ=0–40 with strong Coulomb effects and a spin-independent potential; the n+40Ca benchmark uses the KD02 optical potential with spin-orbit channels.The heavy-ion calculation treats 6Li as a spin-0 projectile.Source: §IV · pp. 7–8

6Li+208Pb benchmark

  • For 6Li+208Pb, phase shifts agree within 0.13° in the transition region (ℓ=25–30), while differences reach about 1.8° for strongly absorbed channels where phase shifts are ill-conditioned.Despite those phase-shift differences, |∆|Sℓ|| remains below 0.002 for strongly absorbed channels, whose elastic-cross-section contribution is negligible.Source: §IV · p. 9
  • 3.3 × 10^-3 is the mean |∆|S|| across all 41 6Li+208Pb partial waves, with ∆|S|≲0.007 in the ℓ=25–30 transition region.PINN-ECS also agrees well with COLOSS for the Rutherford ratio and S-matrix magnitudes.Source: Table III · p. 9, §IV · p. 9
  • Across five matching radii, the extracted S-matrix standard-deviation spread remains below ≈10^-2; separately, apparent matching-radius variation reaches ≈4 × 10^-2 near ℓ≈13, while absolute differences from COLOSS stay below 0.005.Source: §IV · p. 9

n+40Ca benchmark

  • 0.1° phase-shift errors for strongly absorbed n+40Ca channels (ℓ≤4) rise to at most 0.60° across ℓ=5–10; the mean |∆δ| is 0.09° across 20 spin-orbit channels.The comparisons use COLOSS reference values for all 21 channels.Source: §IV · p. 8, Figure 3 · p. 10
  • The n+40Ca anchor scale falls by over three orders of magnitude through ℓ≈6; static anchoring is retained for ℓ≤6 and warm-down engages for ℓ≥7.The reported sensitivity scan finds the converged S-matrix robust to this gating decision.Source: §IV · p. 8, Figure 4 · p. 10

V. DISCUSSION

PINN-ECS reproduces nucleon-nucleus and heavy-ion scattering results, with the driven-equation formulation separating known incident behavior from a scattered wave that decays under ECS. Its applicability is bounded by the scaling radius, and its training cost is substantially higher than conventional integration.

V. DISCUSSION

  • 0.09° mean |∆δ| across all 20 spin-orbit channels, with ∆δ = 0.60° for the worst channel; heavy-ion mean |∆|S|| was 3.3 × 10−3 across 41 waves.For ℓ≤4, nucleon phase-shift errors are at the 0.1° level; for ℓ=25–30, heavy-ion ∆|S| ≲0.007.Source: §V · p. 12
  • ECS damps the scattered wave beyond R0, while the driven equation separates it from the known incident Coulomb wave, simplifying the neural network's task.The source term VshortFℓ also provides an automatic scale for the solution magnitude.Source: §V · p. 12
  • PINN-ECS applies when the relevant reaction physics is confined to radii smaller than R0; long-range electromagnetic transitions at large impact parameters are not directly treated.The passage identifies sub-barrier Coulomb excitation as an example and notes that explicit long-range coupling treatment or semiclassical theory would be needed.Source: §V · p. 12
  • Training dominates computational cost: 20–40 partial waves take roughly one to two hours, versus seconds per partial wave for conventional integration.The passage describes this cost as acceptable for research applications requiring end-to-end differentiability or mesh-free formulations.Source: §V · p. 12

VI. CONCLUSIONS

The conclusions report accurate PINN-ECS results for nucleon-nucleus and heavy-ion scattering, using ECS with a driven-equation formulation. They also describe training innovations and identify inverse and coupled-channel calculations as future directions.

VI. CONCLUSIONS

  • PINN-ECS achieved ∆δ ≲0.1° for n+40Ca channels with ℓ≤4 and mean |∆|S|| = 3.3 × 10−3 across 41 6Li+208Pb partial waves.For all 20 spin-orbit channels in n+40Ca, ∆δ ≤0.60° and mean |∆δ| = 0.09°; the heavy-ion Rutherford ratio reproduces COLOSS over three orders of magnitude.Source: §VI · p. 12
  • Reliability measures include a sigmoid-capped boundary factor, source-dependent anchor scheduling, and Adam followed by L-BFGS training.The anchor warm-down addresses weak-source channels, including the heavy-ion absorption-to-transparency transition and the high-ℓ tail of n+40Ca.Source: §VI · pp. 12–13
  • The proposed future directions include directly fitting optical-potential parameters to data and extending the approach to coupled-channel calculations.The passage also mentions few-body scattering as a direction for future research.Source: §VI · p. 13
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