Differentiable Forward Scattering Model (DFSM)

• Differentiable Forward Scattering Model (DFSM) is a forward operator that accurately represents scattering physics while remaining compatible with gradient-based inversion methods.
• It integrates diverse approaches—such as multislice propagation, sigmoid relaxations, and explicit Jacobians—across applications like coherent surface scattering, radar, and optical tomography.
• DFSM enhances reconstruction accuracy and computational efficiency, enabling effective inverse problem solving in complex physical systems.

A Differentiable Forward Scattering Model (DFSM) is a forward operator for a scattering, diffraction, or wave-propagation problem that is expressed in a form compatible with gradient-based inversion. In the arXiv literature, the term is used for coherent surface scattering imaging, multi-peak Bragg coherent x-ray diffraction imaging, forward-scatter radar, optical diffraction tomography, synthetic aperture radar rendering, photonic scattering-matrix optimization, underwater acoustic localization, weakly perturbed stellar streams, multi-sphere acoustic scattering, and small-angle x-ray scattering analysis (Myint et al., 2022, Maddali et al., 2022, Lei et al., 15 Aug 2025). Across these settings, DFSM denotes a model that preserves the relevant physics of the forward map while enabling derivatives of predicted measurements with respect to object, geometry, material, or latent-field parameters.

1. Scope and defining characteristics

In coherent surface scattering imaging (CSSI), the DFSM addresses grazing-incidence reflection geometry near or below the critical angles of total external reflection, where reflected substrate fields interfere strongly with transmitted fields and generate multiple-scattering effects that a single Fourier transform cannot capture. In that setting, the goal is to map a three-dimensional refractive-index distribution

$n(x,y,z)=1-\delta(x,y,z)+i\,\beta(x,y,z)$

and an incident coherent wave $\phi_0(x,y)$ to the exit wave and then to the far-field intensity (Myint et al., 2022).

In other domains, the same designation is attached to formally different operators. In forward-scatter radar, DFSM replaces a non-differentiable shadow indicator by a continuous Secondary Wave-Source Response Field (SWRF), turning Kirchhoff–Fresnel diffraction into a differentiable inverse problem. In multi-peak BCDI, it couples several Bragg reflections into a single optimization. In optical diffraction tomography, it is the exact Lippmann–Schwinger map together with an explicit Jacobian. In photonics, it is a differentiable scattering matrix $S(\lambda,p)$ computed without differentiating eigenvectors. In underwater acoustics, it is a modular multipath superposition model. In SAXS analysis, it is a differentiable composition of a surrogate monodisperse scattering model and a trainable polydispersity layer (Lei et al., 15 Aug 2025, Maddali et al., 2022, Soubies et al., 2017, Zhu et al., 2020, Kari et al., 30 Mar 2025, Bånkestad et al., 22 May 2026).

This suggests a recurring template: physical parameters are mapped to a predicted field or intensity, a loss is defined against measured data, and gradients are propagated through the entire pipeline. The common feature is not a single governing equation, but the requirement that the forward scattering computation remain faithful enough to the underlying physics while also remaining differentiable.

2. Mathematical structure of representative DFSMs

The CSSI formulation derived from the multislice formalism discretizes the sample along the surface-normal $Z$ into $N$ slices of thickness $\Delta z$. For slice $j$,

$$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$

and the transmission operator is

$$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$

Free-space propagation between slices is applied in Fourier space with

$$P(k_x,k_y;\Delta z)=\exp\!\left[-i\,\Delta z\,(k_x^2+k_y^2)/(2k_0)\right],$$

leading to the recursive update

$\phi_0(x,y)$0

and the predicted far-field intensity

$\phi_0(x,y)$1

The details state that the notation follows Myint et al. (2022), with explicit forward-differentiable implementation details added (Myint et al., 2022).

A different mathematical strategy appears in forward-scatter radar. There, the classical Kirchhoff–Fresnel diffraction model contains a binary indicator $\phi_0(x,y)$2 for the shadow region, which is non-differentiable at the target boundary. The DFSM replaces $\phi_0(x,y)$3 by $\phi_0(x,y)$4, where

$\phi_0(x,y)$5

The field prediction becomes

$\phi_0(x,y)$6

In this formulation, all physics factors are known and the only unknown is the continuous field $\phi_0(x,y)$7 (Lei et al., 15 Aug 2025).

Other DFSMs preserve differentiability without resorting to a sigmoid relaxation. In optical diffraction tomography, the forward model is the exact Lippmann–Schwinger equation,

$\phi_0(x,y)$8

with discrete form

$\phi_0(x,y)$9

and an explicit Jacobian

$S(\lambda,p)$0

In photonics, the forward map is the scattering matrix

$S(\lambda,p)$1

and differentiation is performed through auxiliary matrices $S(\lambda,p)$2 and a Lyapunov-type equation for $S(\lambda,p)$3, thereby sidestepping explicit differentiation of eigenvectors (Soubies et al., 2017, Zhu et al., 2020).

3. Mechanisms for maintaining differentiability

The cited literature uses several distinct mechanisms to keep the forward model differentiable.

Domain	Forward model	Differentiability mechanism
CSSI	Multislice propagation	PyTorch FFTs, complex tensors, exponentials
Forward-scatter radar	FKDM with SWRF	Sigmoid relaxation of the shadow indicator
Optical diffraction tomography	Lippmann–Schwinger	Explicit Jacobian and adjoint linear solve
Photonics	Scattering matrix via RCWA	Eigenvector-free derivative algorithm
SAXS	Surrogate plus quadrature	Automatic differentiation through network and inverse-CDF mapping

In CSSI, each slice parameter $S(\lambda,p)$4 and $S(\lambda,p)$5 is a PyTorch tensor with requires_grad=True, and the forward pass is written entirely in PyTorch using torch.complex64, torch.fft.fft2, torch.fft.ifft2, and element-wise exponentials and multiplications. In multi-peak BCDI, every step from object amplitude and displacement to detector intensity is composed of multiplication, $S(\lambda,p)$6, FFT, modulus, and square, so autodiff engines back-propagate the loss automatically. In SAXS, differentiability runs through the neural surrogate, the Gauss–Legendre quadrature layer, and the inverse-CDF mapping for a truncated log-normal radius distribution (Myint et al., 2022, Maddali et al., 2022, Bånkestad et al., 22 May 2026).

Some DFSMs rely instead on explicit derivatives. The optical diffraction tomography model computes the gradient of the data-fidelity term using two linear solves per illumination and does not require back-propagation through the forward-iteration history. The photonics formulation computes $S(\lambda,p)$7 by differentiating $S(\lambda,p)$8 into a Lyapunov-type equation, then obtaining $S(\lambda,p)$9 from the upper-right block of a block-matrix exponential. The differentiable SAR renderer of Fu & Xu (2023) derives first-order gradients analytically through soft rasterization, soft occlusion, and Gaussian slant-range mapping (Soubies et al., 2017, Zhu et al., 2020, Fu et al., 2022).

A common misconception is that differentiability requires simplification to weak-scattering or purely Fourier-transform models. The literature does not support that view. The CSSI model is explicitly introduced because conventional CDI reconstruction techniques cannot be directly applied near the critical angle, and the ODT model is built from the exact Lippmann–Schwinger formulation rather than the Born or Rytov approximations (Myint et al., 2022, Soubies et al., 2017).

4. Losses, regularization, and inverse-problem formulations

DFSMs are typically embedded in variational or gradient-based inverse problems. In CSSI, a representative loss is the mean-squared error on amplitudes,

$Z$0

or, with regularization,

$Z$1

where $Z$2 may be a support constraint or total variation. Parameters are updated by an optimizer such as Adam, with optional shrink-wrap support (Myint et al., 2022).

In forward-scatter radar, the total cost is

$Z$3

The data term is a log-Huber loss applied to complex residuals, the entropy term counteracts premature sigmoid saturation, and the geometric prior is decomposed into smoothness, connectivity, and compactness. After optimization, $Z$4 is thresholded at $Z$5 to obtain a hard binary profile (Lei et al., 15 Aug 2025).

Other DFSMs follow analogous patterns with domain-specific objectives. Multi-peak BCDI minimizes a mean-square-root-amplitude error over all Bragg reflections plus a small TV term. Underwater acoustic localization defines

$Z$6

so that source position and network weights can be jointly adapted at inference time. The differentiable SAR renderer optimizes a composite objective

$Z$7

combining silhouette IoU, SAR texture fidelity, Laplacian smoothness, and planarity. In point-scattering radar models, coherent, noncoherent, and sequential losses are compared for fitting range profiles (Maddali et al., 2022, Kari et al., 30 Mar 2025, Fu et al., 2022, Chance et al., 2022).

These formulations show that DFSM is not tied to a single regularizer or optimizer. What recurs is the use of differentiability to make physically structured inverse problems accessible to first-order methods.

5. Implementations and computational properties

Implementation practice depends strongly on the forward operator. The CSSI DFSM keeps the refractive-index volume in GPU memory, uses cuFFT under torch.fft, avoids host–device transfers inside the slice loop, and benefits from ATen/CUDA kernel fusion. The details report that, on a single NVIDIA A100, a $Z$8 sample with $Z$9 slices and $N$0 lateral voxels can be propagated through $N$1 slices in $N$2, while the backward pass incurs roughly a $N$3 slowdown (Myint et al., 2022).

Some DFSMs are designed primarily to reduce memory or evaluation cost. In optical diffraction tomography, the explicit Jacobian reduces peak memory from $N$4 to $N$5 because it eliminates the need to store the entire forward-iteration history. In the SAXS framework, the full simulator for $N$6 GRFs and $N$7 radii costs approximately $N$8 per evaluation, whereas the surrogate plus differentiable quadrature costs approximately $N$9, yielding an approximately $\Delta z$0 speedup and making $\Delta z$1 AdamW restarts feasible in $\Delta z$2 on a single GPU (Soubies et al., 2017, Bånkestad et al., 22 May 2026).

JAX-based DFSMs emphasize whole-pipeline differentiability. The stellar-stream model uses pure JAX together with galax and diffrax, so jax.grad can be applied directly to statistics of the density and kinematic power spectra. After a one-time compilation cost of approximately $\Delta z$3, each simulation scales linearly in the number of kicks; for the forecast setup with $\Delta z$4, the cost is approximately $\Delta z$5 per kick, and for $\Delta z$6 on a $\Delta z$7 grid the memory usage is approximately $\Delta z$8 of GPU RAM. The differentiable SAR renderer reports forward times of approximately $\Delta z$9 for approximately $j$0 facets and approximately $j$1 for approximately $j$2 facets on $j$3 grids, with additional backward runtimes reported for T-72, Envisat, and Tiangong-1 reconstructions (Montel et al., 8 Jun 2026, Fu et al., 2022).

6. Reconstructions, benchmarks, and limitations

The empirical role of DFSM is clearest in reconstruction and forecast results. In CSSI, a simulated buried-chip example with $j$4 voxels is reported to recover a $j$5 gold-in-silicon pattern in approximately $j$6 epochs using amplitude MSE alone, and in a FIB-deposited pillar example with strong AFM priors and optimization of $j$7 voxels, the top-surface taper angle is refined by $j$8 epochs to reproduce the key beating fringes in the single-shot image (Myint et al., 2022).

In forward-scatter radar, convex polygons are evaluated in both Transition Zone and Far-Field geometries using Intersection-over-Union and $j$9 Hausdorff Distance. Under noise-free signals, Transition Zone IoU lies in $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$0 and $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$1HD in $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$2, while Far-Field IoU lies in $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$3 and $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$4HD in $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$5. For non-convex targets under severe mixed noise at $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$6 with AWGN plus $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$7 impulse noise, Transition Zone IoU is approximately $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$8 with $$\delta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\delta(x,y,z)\,dz,\qquad \beta_j(x,y)=\int_{z_j}^{z_j+\Delta z}\beta(x,y,z)\,dz,$$9HD $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$0, and Far-Field IoU is approximately $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$1 with $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$2HD $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$3 (Lei et al., 15 Aug 2025).

In multi-peak BCDI, a dislocation-free synthetic crystal reconstructed from five Bragg peaks reaches a final spatial resolution of approximately $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$4, whereas conventional single-peak phase retrieval under identical noise gives approximately $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$5. A synthetic crystal with two screw dislocations is reported to recover the two orthogonal trenches and half-plane phase jumps, and an experimental SiC nanoparticle reconstruction yields a clean D-shaped cross section with smooth, continuous displacement fields (Maddali et al., 2022).

Several papers also delimit where DFSM-based inversion becomes difficult. In underwater acoustics, joint DFSM adaptation cuts RMSE by $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$6–$$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$7 for depth offsets up to $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$8, but for large offsets the small-mismatch assumptions break and adaptation stalls. In point-scattering radar characterization, coherent-only fitting often converges to a wrong local minimum when initialization is inaccurate, while sequential noncoherent-to-coherent fitting mitigates but does not eliminate local-minimum sensitivity. In the GD-1-like stellar-stream forecasts, residual small-scale excess power is attributed to full $$T_j(x,y)=\exp[i\,k_0\,\delta_j(x,y)-k_0\,\beta_j(x,y)].$$9 effects not captured by the $$P(k_x,k_y;\Delta z)=\exp\!\left[-i\,\Delta z\,(k_x^2+k_y^2)/(2k_0)\right],$$0 theory, even though kinematic information tightens constraints on the free-streaming cutoff scale by a factor of approximately $$P(k_x,k_y;\Delta z)=\exp\!\left[-i\,\Delta z\,(k_x^2+k_y^2)/(2k_0)\right],$$1–$$P(k_x,k_y;\Delta z)=\exp\!\left[-i\,\Delta z\,(k_x^2+k_y^2)/(2k_0)\right],$$2 relative to density alone (Kari et al., 30 Mar 2025, Chance et al., 2022, Montel et al., 8 Jun 2026).

Taken together, these results show that DFSM is best understood as a methodological class rather than a single model: a physics-structured forward operator made differentiable so that inverse scattering, reconstruction, localization, and forecast problems can be solved by gradient-based optimization. The specific operator may be multislice propagation, an integral equation, a scattering matrix, a differentiable renderer, or a surrogate-augmented ensemble average; the unifying principle is the retention of physically meaningful forward structure under differentiation.