Limits of the inverse scattering problem

Abstract: The main goal of tomography is the reconstruction of density function out of its line integrals (integral measurements of this density along x-ray lines). Such construction is possible and known as inverse x-ray/Radon transformations. We are interested in generalization of this problem to the case of particles. For this we need to limit the speed of these particles, otherwise the problem is reduced to the previous one. The question we discuss in this paper is how low can this speed be depending on the parameters of the studied potential. We study this problem using simple theoretical examples and Machine Learning pipeline to restore the potential.

• The paper shows that reconstruction requires initial kinetic energy to exceed the characteristic interior-to-boundary potential variation, a threshold confirmed for harmonic and constant-force potentials.
• The authors develop finite-velocity perturbation expansions and propose single-speed iteration and multi-speed extrapolation to correct deviations from the infinite-velocity Radon transform.
• Machine-learning tests show reconstruction errors rise sharply when kinetic and potential energies become comparable, while residence time strongly encodes field amplitude and Radon methods preserve shape but lose scale.

Problem statement and main claim

The paper addresses the classical inverse scattering problem for Newtonian particles in two dimensions, generalizing X-ray tomography. In the infinite-velocity limit the particle trajectories are straight chords, the measured momentum change reduces to the vector Radon transform of the force field $\mathbf{J}_\infty(L)=\int_L \mathbf{F}\,ds$, and standard filtered back-projection reconstructs $\mathbf{F}$ exactly. The central question is how far below this limit one can go before reconstruction fails. The authors' principal claim is that the minimal kinetic energy required for reconstruction equals the magnitude of the potential-energy difference between the interior and the boundary of the domain,

$E = \max(|U_{\text{in}} - U_{\text{out}}|),$

so that reconstruction quality is governed by the ratio between initial kinetic energy and this characteristic potential variation. This claim is supported by two exactly solvable examples and by a machine-learning pipeline applied to synthetic random fields.

Perturbative expansion at finite velocity

The authors derive a systematic expansion of the finite-speed vector ray integral in powers of the dimensionless parameter $\epsilon = \delta U/(mv^2)$:

$$\mathbf{J}_v(L) = \mathbf{J}_\infty(L) + \epsilon\,\mathbf{Q}[\mathbf{F}](L) + \epsilon^2\,\mathbf{P}[\mathbf{F}](L) + O(\epsilon^3),$$

where $\mathbf{Q}$ and $\mathbf{P}$ are quadratic and cubic functionals of $\mathbf{F}$ and its derivatives, integrated along the unperturbed chord. The derivation uses fixed boundary endpoints, a normalized time variable, and pinned-end boundary-value problems for successive trajectory corrections; flight times enter at odd half-integer powers of $\epsilon$ and deflections at integer powers. Applying the linear inverse Radon operator yields a naively reconstructed field with explicit correction terms, which can be inverted either by single-speed iteration (converging rapidly for $v > v_c$) or by multi-speed extrapolation: measurements at two or more speeds allow algebraic elimination of the first-order correction, or regression in $\mathbf{F}$0 extrapolated to zero — an RG-flow-like procedure requiring no prior knowledge of the correction functionals.

Exactly solvable examples

For the harmonic potential $\mathbf{F}$1 on the unit disk, the exact scattering relation $\mathbf{F}$2 exhibits a sharp topological transition at $\mathbf{F}$3: below $\mathbf{F}$4 the map from injection angle to exit point is two-to-one onto a restricted arc; above $\mathbf{F}$5 it becomes a bijection covering the full circle. The exact ray integral is

$\mathbf{F}$6

whose series converges absolutely precisely when $\mathbf{F}$7. Notably, all odd powers vanish ($\mathbf{F}$8), and the explicit inverse Radon transform of the cubic term reproduces exactly the perturbative prediction, confirming internal consistency. Reconstruction then requires $\mathbf{F}$9, matching the claimed energy criterion.

For a constant force, the exact closed-form ray integral yields an expansion whose convergence condition $E = \max(|U_{\text{in}} - U_{\text{out}}|),$0 again produces a critical velocity set by the kinetic-to-potential ratio. From both examples the authors conjecture a universal nonlinear relation between $E = \max(|U_{\text{in}} - U_{\text{out}}|),$1 and $E = \max(|U_{\text{in}} - U_{\text{out}}|),$2; it is exact for these two cases but only approximate for general potentials, where non-local effects are not controlled.

Machine-learning reconstruction

The numerical pipeline combines a CUDA RK4 forward simulator on a $E = \max(|U_{\text{in}} - U_{\text{out}}|),$3 grid with a Force-Field Prediction Model (FFPM): a Johnson-style encoder–residual-bottleneck–decoder generator (8.1M parameters, six residual blocks chosen over DiT variants on time-to-quality grounds) trained adversarially with two PatchGAN discriminators, an $E = \max(|U_{\text{in}} - U_{\text{out}}|),$4 reconstruction loss, and a field-domain NCE contrastive term against a Radon prior. Input consists of three boundary-measurement channels (exit direction, speed change, residence time) concatenated with the Radon reconstruction; the model predicts the force field directly, avoiding the additive-constant ambiguity of the potential.

An information analysis shows that the residence-time channel dominates the encoding of field amplitude (Pearson correlation $E = \max(|U_{\text{in}} - U_{\text{out}}|),$5 with $E = \max(|U_{\text{in}} - U_{\text{out}}|),$6, versus $E = \max(|U_{\text{in}} - U_{\text{out}}|),$7 for the other channels), and an auxiliary Img2Vec regressor recovers $E = \max(|U_{\text{in}} - U_{\text{out}}|),$8 over twelve decades from raw measurements alone.

Specialist models trained at sixteen amplitudes reveal a pronounced error increase for $E = \max(|U_{\text{in}} - U_{\text{out}}|),$9, i.e. $\epsilon = \delta U/(mv^2)$0 — the interval containing the harmonic critical value $\epsilon = \delta U/(mv^2)$1. An out-of-distribution test on a centered paraboloid shows degradation beginning in the same energy region, supporting the interpretation that the transition reflects loss of information content in the forward operator rather than ordinary distribution shift. The two methods fail differently: Radon reconstruction loses amplitude while preserving shape (correlation stays above 0.98 even where relMAE exceeds 0.9), whereas FFPM retains extrema but introduces noise and symmetry breaking — implying that integral metrics alone do not characterize physical reconstruction quality.

Limitations and open questions

The paper concedes several restrictions. The universal relation between $\epsilon = \delta U/(mv^2)$2 and $\epsilon = \delta U/(mv^2)$3 is conjectural outside the two solvable cases. The naive accessibility diagnostic ignores connectivity of reachable regions, so energetically accessible pockets behind inaccessible barriers are miscounted. All experiments use noiseless synthetic data at a single launch speed per configuration; robustness to measurement noise, missing trajectories, and independent test sets with repeated training runs are explicitly deferred to future work. Whether the energy criterion $\epsilon = \delta U/(mv^2)$4 holds beyond smooth conservative fields — e.g., for vortical forces or frictional dynamics — remains open, as does a rigorous proof connecting the observed $\epsilon = \delta U/(mv^2)$5 breakdown threshold to the perturbative series divergence.

Conclusion

The paper establishes, through exact solutions and consistent numerical evidence, that the applicability of Newtonian-particle tomography is bounded by an energy criterion: reconstruction succeeds when the initial kinetic energy exceeds the characteristic interior-to-boundary potential variation, and degrades sharply as these scales become comparable. The combination of a perturbative multi-speed extrapolation scheme and a learned reconstructor provides complementary practical routes to finite-speed inversion within that regime.