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Simulator-Refined Diffusion for Radio-Frequency Inverse Design

Jinhao Liang, Jacob K. Christopher, Michael Frei, Tommaso Dreossi, Nando Fioretto

arXiv:2609.38363v1cs.LGcs.AI

TL;DR

RF PCB inverse design must match target electromagnetic responses, but full-wave verification is expensive and surrogate-based guidance can be unreliable. SRD uses surrogate gradients to propose edits and simulator feedback to select updates, reporting lower simulated S-parameter error than conditional diffusion for both in-distribution and constructed out-of-distribution targets.

  • Problem

    RF inverse design requires matching target S-parameters despite expensive full-wave evaluations and surrogate guidance that may not track simulated loss.

    Source: §1 · pp. 1–2, §3 · p. 4

  • Method

    SRD uses a differentiable surrogate to propose geometric edit directions and a full-wave simulator to assess and select candidate updates within a call budget.

    Source: §5.2 · p. 6

  • Results

    Up to 21.2% and 19.8% reductions in simulated S-parameter loss are reported for in-distribution and constructed out-of-distribution targets, respectively, relative to conditional diffusion.

    Source: §1 · p. 2, §7 · p. 10

  • Takeaways & Limitations

    Across the reported targets, SRD combines targeted edits with simulator selection and improves simulated response matching over the conditional diffusion baseline.

    Source: §1 · p. 2, §7 · p. 10

    Simulation error remains possible, so the method is not sufficient validation for fabrication or safety-critical deployment and requires physical validation.

    Source: ETHICS STATEMENT · p. 11

Abstract

from arXiv · show

Diffusion models have shown potential in inverse design of printed circuit boards (PCBs), enabling the generation of layouts conditioned on target S-parameters. Despite this promise, applying diffusion models to PCB layout generation remains challenging due to their difficulty in meeting the quantitative electromagnetic specifications. A common approach is gradient-based guidance, which biases the diffusion sampling process with the gradient of an objective used for evaluation. However, full-wave electromagnetic simulators are accurate but expensive and typically non-differentiable, whereas differentiable surrogates are informative but not always reliable. To address these limitations, this paper proposes Simulator-Refined Diffusion (SRD), a novel combination of a low-fidelity differentiable surrogate and a high-fidelity non-differentiable simulator within the diffusion sampling process. Unlike standard zeroth-order optimization, which requires a great number of random perturbations, our approach uses the surrogate's gradient to propose the perturbation direction while the simulator then searches based on this direction to identify an effective design update. Experimental results across different settings show that this method consistently outperforms current state-of-the-art methods, producing layouts whose simulated S-parameters match the target specifications up to 21.2% closer for in-distribution targets and up to 19.8% for out-of-distribution targets.

1 INTRODUCTION

SRD addresses the mismatch between surrogate-guided diffusion and simulated RF performance by using surrogate gradients to propose edits and a full-wave simulator to select them. Across in-distribution and constructed out-of-distribution targets, SRD reduces simulated S-parameter error relative to conditional diffusion.

1 INTRODUCTION

  • RF PCB inverse design seeks a physical geometry matching target S-parameters, but the inverse mapping is ill-posed and full-wave verification is expensive.Small geometric changes can alter responses across frequency, while each high-fidelity evaluation requires a full-wave solution.Source: §1 · p. 1
  • Surrogate guidance alone can yield negligible improvement over unguided diffusion because low predicted loss can coexist with large simulated error.The paper notes low correlation between surrogate and simulator losses, especially in low-loss and out-of-distribution regions.Source: §1 · p. 2, Figure 1 · p. 2
  • SRD uses surrogate gradients to propose geometric edits, then uses full-wave simulation to assess and select candidate updates during diffusion sampling.The simulator budget is allocated to signed displacements along the proposed direction or to alternative directions based on the initial probe.Source: §1 · p. 2
  • Up to 21.2% and 19.8% reductions in simulated S-parameter error are reported for in-distribution and constructed out-of-distribution targets, respectively, versus conditional diffusion.The evaluation spans 250 real targets across 25 RF layout templates and 50 constructed out-of-distribution targets.Source: §1 · p. 2

2 RELATED WORK

Prior work applies optimization and generative models to electromagnetic inverse design, and diffusion guidance adapts pretrained models using task objectives. SRD differs by using a surrogate to propose local geometric directions and reserving expensive simulator calls for selecting edits.

Electromagnetic and RF inverse design

  • Conditional diffusion has been applied to EM structures and PCB inverse design, while SRD uses a surrogate-proposed geometric direction and a small simulator-call budget to select a signed displacement.The cited PCB diffusion work uses surrogate-based ranking to improve response matching (Dreossi et al., 2026).Source: §2 · p. 3

Diffusion guidance for inverse problems

  • Diffusion guidance methods modify samples or use measurement-consistency gradients, but their appeal depends on reliable guidance models; SRD targets expensive, non-differentiable simulation with a surrogate used mainly to propose search directions.Examples include SDEdit, diffusion posterior sampling, and training-free guidance.Source: §2 · p. 3

Inference-time search and differentiable methods

  • Best-of-n selects among whole trajectories, tree search reuses rollout feedback, and zeroth-order methods sample directions; SRD instead refines one trajectory along a surrogate-proposed direction.The passage notes that tree search and many sampled directions can be costly when rewards require full-wave simulations.Source: §2 · p. 3

3 SETTINGS AND GOALS

The paper formulates RF PCB inverse design as finding an admissible layout whose simulated response matches a target under a finite full-wave simulation budget. The formulation separates the expensive simulator objective from its inexpensive surrogate approximation.

3 SETTINGS AND GOALS

  • A layout and fixed context determine frequency-dependent S-parameters, and inverse design seeks an admissible layout matching a specified response.The context includes port positions, substrate permittivity, and thickness; admissibility requires grid representability, containment within the board outline, and simulator evaluability.Source: §3 · p. 3
  • The full-wave simulator is the reference forward model, while a neural surrogate provides an inexpensive differentiable approximation.Computing the response requires solving Maxwell’s equations, for example by finite elements or the method of moments.Source: §3 · p. 4
  • The simulator loss measures design quality against the target, and the surrogate loss approximates that objective but may be biased.The two losses are defined by applying a response distance to the simulator or surrogate output and the target response.Source: §3 · p. 4
  • Given a target response, fixed board context, and simulator-call budget, the goal is to find an admissible layout with the best simulated match to the target.The budget constrains the total number of full-wave simulator calls used to generate, select, and evaluate a returned layout.Source: §3 · p. 4

4 GUIDED DIFFUSION FOR INVERSE DESIGN

Conditional diffusion reverses a corruption process to generate layouts conditioned on target response and board context. Loss guidance adds an inexpensive surrogate gradient, but differentiability alone does not ensure better simulated responses.

4 GUIDED DIFFUSION FOR INVERSE DESIGN

  • Conditional diffusion reverses a gradual corruption process, using the target response and board context to guide noisy states toward layouts.At each step, the model produces a clean-layout prediction that can be evaluated for objectives and edits during sampling.Source: §4 · p. 4
  • Guided diffusion adds a target-dependent loss gradient through the differentiable surrogate because the full-wave simulator does not expose that gradient.The surrogate loss is extended to the continuous layout representation, with a differentiable decoder when needed.Source: §4 · p. 4
  • Low surrogate loss can coexist with substantially larger simulator loss, so differentiability alone does not establish accurate local directions or reliable step sizes.Optimizing surrogate loss may favor layouts that do not improve simulator loss, particularly when edits leave the surrogate’s training distribution.Source: §4 · pp. 4–5

5 SIMULATOR-REFINED DIFFUSION

Simulator-Refined Diffusion (SRD) uses surrogate gradients to propose edits to a structured layout, then uses full-wave simulator evaluations to select updates within a limited call budget. Its search operates on rectangle geometries and adapts candidate generation based on whether an initial trial improves simulator loss.

5 SIMULATOR-REFINED DIFFUSION

  • SRD uses surrogate gradients to propose geometric edits and full-wave simulator evaluations to select the candidate with the lowest simulator loss.A trial displacement determines whether the remaining candidates explore step sizes along the proposed direction or alternative directions.Source: §5 · p. 5

5.1 STRUCTURED EDIT SPACE

  • In the structured edit space, each rectangle is represented by its center coordinates, width, and height, so edits directly control geometry in R4n.Pixel-space changes do not directly control the resulting rectangle changes through the learned geometry map.Source: §5.1 · p. 5

5.2 SURROGATE-PROPOSED, SIMULATOR-SELECTED EDITS

  • A differentiable surrogate proposes a descent direction, but its predicted improvement may not reduce simulator loss, especially when edits move geometry away from the training distribution.SRD first tests a fixed-step displacement with the simulator before deciding how to generate the remaining candidates.Source: §5.2 · p. 6
  • The directional variant tests further step sizes along the proposed direction after a successful probe, or along its opposite after an unsuccessful probe.It remains confined to the surrogate-defined axis, and a failed probe does not guarantee improvement in the opposite direction.Source: §5.2 · p. 6
  • The adaptive variant switches to random rectangle-parameter perturbations after an ineffective probe, while the hybrid variant combines directional candidates with random perturbations at each refinement step.The hybrid broadens the search but requires a larger simulator-call budget.Source: §5.2 · p. 6
  • Including the unedited geometry among candidates ensures each refinement has simulator loss no greater than its starting geometry.Refinement timing and candidate-set size trade off geometry stability, opportunities for later denoising to overwrite edits, and simulator-call cost.Source: §5.2 · p. 6

6 EXPERIMENTS

Across the test and OOD sets, SRD reduces simulated S-parameter loss more than the baselines, combining surrogate-proposed edits with simulator-based selection. Ablations and template-level results show how performance varies with the selection strategy and target difficulty.

6 EXPERIMENTS

  • Evaluation covers 250 test samples across 25 templates and 50 constructed OOD targets, scoring response error over 51 frequency points from 1–20 GHz.All methods share the same frozen diffusion and surrogate models, trained on approximately 1.26M procedurally generated PCBs.Source: §6 · p. 7

6.1 COMPARISON ACROSS METHODS

  • With matched Palace-call budgets, SRD raises loss reductions versus CD to 19.06% on the test set and 17.98% on the OOD set; all SRD variants beat baselines on mean and median loss.The comparisons use simulated S-parameter loss Lsim; Table 1 also reports Palace-call budgets and simulation success.Source: §6.1 · p. 7, Table 1 · p. 8
  • On the test set, SRD (hybrid) has the lowest mean loss; OOD variants perform similarly, with SRD (directional) lowest, while probe spending leaves fewer calls for larger perturbations.The comparison uses the methods’ guidance-step and Palace-call budgets described in Table 1.Source: §6.1 · p. 8, Table 1 · p. 8
  • SRD’s gains over CD are most pronounced on medium and hard templates, while easy templates also improve but have less room for improvement.Figure 3 groups test templates by difficulty; lower S-parameter loss is better.Source: Figure 3 · p. 8, §6.1 · p. 8
  • A shown SRD layout has target-closer responses despite shifting metal geometry relative to fixed ports, illustrating novel layouts for a design objective.Figure 4 places the target responses at left, the reference layout at center, and the SRD layout at right.Source: Figure 4 · p. 8, §6.1 · pp. 8–9

6.3 ABLATIONS

  • With random perturbations, Palace-based selection lowers mean loss to 0.2819, while surrogate-based selection gives 0.3263, slightly above CD at 0.3222.The ablation compares candidate selection using simulator feedback versus surrogate predictions.Source: §6.3 · p. 9, Table 2 · p. 9
  • Under surrogate-based selection, descent directions yield mean loss 0.2873 versus 0.3263 for random perturbations; SRD (adaptive) achieves the ablation’s lowest mean and median losses, 0.2608 and 0.1977.SRD (adaptive) combines surrogate-proposed directions, Palace-based selection, and a conditional fallback to random perturbations.Source: §6.3 · p. 9, Table 2 · p. 9

7 LIMITATIONS AND CONCLUSION

SRD combines surrogate-proposed edit directions with full-wave simulator feedback, improving simulated S-parameter matching across in-distribution and constructed out-of-distribution targets. The paper identifies limits in geometric representation and search directions, and cautions that simulation is not sufficient validation for fabrication or safety-critical deployment.

7 LIMITATIONS AND CONCLUSION

  • SRD variants reduce mean simulated S-parameter loss by up to 21.2% and 19.8% on in-distribution and constructed out-of-distribution targets, respectively, versus a state-of-the-art conditional diffusion model.The evaluation spans 250 test targets across 25 layout templates and 50 constructed out-of-distribution targets.Source: §7 · p. 10
  • At selected denoising steps, SRD uses surrogate gradients to propose edit directions and full-wave simulator probes to guide candidate edits.An ablation study supported complementary roles for surrogate directions and simulator-based selection.Source: §7 · p. 10
  • Future work could extend SRD’s rectangle-based representation to finer geometric features and more complex topologies, and explore directions beyond those proposed by the surrogate.Source: §7 · p. 10
  • Simulation error remains a concern, so the method is not sufficient validation for fabrication or safety-critical deployment; downstream use requires domain review, manufacturing checks, regulatory compliance where applicable, and physical validation.Source: ETHICS STATEMENT · p. 11

A.1 SIMULATOR CONFIGURATION AND EVALUATION

The appendix specifies how candidate layouts are converted into solid models, meshed, and evaluated, and records the board, port, boundary, and solver settings used for simulation.

A.1 SIMULATOR CONFIGURATION AND EVALUATION

  • Candidate and final layouts are converted into solid models, meshed with gmsh, and evaluated with Palace, which returns the complete S-matrix at 51 frequencies from 1 to 20 GHz.Internal frequency-sweep solves count as part of each Palace invocation.Source: §A.1 · p. 13
  • The simulated board is an 8 × 8 mm, 64 × 64-raster layout with 35 µm copper signal and ground layers; the substrate properties are listed in Table 3.Vias have a 0.3 mm diameter and a 0.6 mm pad diameter.Source: §A.1 · p. 13, Table 3 · p. 13
  • Each feed uses a 50 Ω lumped port, while the air box extends 2 mm laterally and above the board and uses absorbing conditions on its five side and top faces.The bottom face has no explicit boundary tag and uses the solver’s natural perfect-magnetic-conductor condition.Source: §A.1 · p. 13
  • Meshes use first-order tetrahedral geometry without adaptive refinement, and Palace uses order-2 N´ed´elec elements, SuperLU DIST, and an adaptive reduced-order frequency sweep.Each solve uses one MPI rank on an NVIDIA A10G GPU with a 900 s timeout.Source: §A.1 · p. 13

A.2 COMPLETE SAMPLING ALGORITHM

SRD refines selected late-stage diffusion predictions by proposing geometric edits from a surrogate gradient, testing them with the full-wave simulator, and selecting the lowest-loss candidate within the call budget.

A.2 COMPLETE SAMPLING ALGORITHM

  • The algorithm takes the target and context, reverse-sampling steps, a refinement window, step size, improvement threshold, simulator-call budget, variant, and candidate counts as inputs.It returns a decoded PCB layout.Source: §A.2 · p. 14
  • When refinement is permitted, SRD predicts a clean sample, extracts its geometry, and uses the negative surrogate-loss gradient to propose an edit direction.The initial probe compares the baseline and a trial edit at step size h.Source: §A.2 · p. 14
  • The initial probe determines whether the adaptive variant explores random perturbations or continues directionally; the hybrid variant adds random candidates to its directional search.Source: §A.2 · p. 14
  • Directional candidates vary step size along the supplied direction, while random candidates use configured perturbations; the algorithm simulates candidates and selects the lowest-loss layout.Selection includes the baseline and initial probe.Source: §A.2 · p. 14

A.3 GEOMETRIC PERTURBATIONS

The appendix shows how SRD changes layouts, how those changes relate to proximity to the diffusion prior, and how generated responses and variant geometries compare qualitatively.

A.3 GEOMETRIC PERTURBATIONS

  • From a shared layout at step 17, SRD adds metal along existing boundaries and widens the conducting strip at steps 18–19, while CD shows no visible change; port locations remain fixed.Figure 6 shows intermediate layouts from the 20-step sampling process.Source: §A.3 · p. 14, Figure 6 · p. 15

A.4 EFFECT ON THE DIFFUSION PRIOR

  • SRD retains training-set distances close to CD for many samples while achieving the lowest mean and median S-parameter losses among four methods, suggesting improved response accuracy with limited changes to the learned distribution.Figure 7 compares nearest-neighbor distances in DINOv3 embedding space against CD; the comparison is read alongside Table 1.Source: §A.4 · p. 14, Figure 7 · p. 15

A.5 QUALITATIVE DESIGN COMPARISON

  • In an example, SRD more closely matches both reflection and transmission responses across the frequency range; CD has a transmission dip near 15 GHz absent from the target, while GGD and DPO retain reflection deviations.Figure 8 presents PCB layouts and their S-parameter magnitude responses.Source: §A.5 · p. 16, Figure 8 · p. 16
  • Adaptive and directional SRD often produce similar overall structures with differences in conductor widths and local branch geometry, while hybrid layouts show more pronounced differences in several cases.Figure 9 shows 25 layouts for each SRD variant.Source: §A.5 · p. 16, Figure 9 · p. 16

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