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AcoustiPrint: Physics-Guided Neural Design of Manufacturable Acoustic Structures

Shanmuka Sadhu, Harshvardhan C. Takawale, Yehia Zakaria, Ahmed Allam, Nirupam Roy

arXiv:2610.30411v1cs.LGcs.AI

TL;DR

AcoustiPrint tackles the difficulty of retaining expressive neural inverse design while satisfying discrete fabrication constraints. It places a printability operator and surrogate learning mechanisms in the design pipeline, and produces printable acoustic structures with improved focal accuracy and acoustic energy concentration over phase- and density-based baselines.

  • Problem

    Neural inverse-design methods can restrict the design space, use soft manufacturability penalties, or post-process structures, leaving a mismatch between simulated and fabricated geometries.

    Source: Introduction · p. 1

  • Method

    AcoustiPrint uses an implicit neural representation and printer-parameterized printability operator, with surrogate gradients and a self-supervised objective for learning through discrete constraints.

    Source: Introduction · p. 1, Differentiable Printability Operators · p. 4, Surrogate Backpropagation · p. 5

  • Results

    AcoustiPrint produces directly printable acoustic structures with improved focal accuracy and acoustic energy concentration over phase- and density-based baselines.

    Source: Introduction · p. 2

  • Takeaways & Limitations

    The paper presents a methodology for neural design combining non-differentiable manufacturing constraints with differentiable physics.

    Source: Conclusion · p. 7

    The printability operator covers layered additive processes with a fixed build direction, not non-planar or multi-axis deposition.

    Source: Differentiable Printability Operators · p. 4

Abstract

from arXiv · show

Recent work in additive manufacturing and neural inverse design enables the creation of complex three-dimensional structures to manipulate physical fields, yet these designs must also adhere to discrete, non-differentiable fabrication constraints. Current methods limit the design space by imposing differentiable soft manufacturability loss penalties, or by fixing structures through post-processing, causing a discrepancy between the geometry optimized by the physics model and the final fabricated result. Thus, we present AcoustiPrint, a physics-informed neural architecture for constrained inverse design that represents three-dimensional geometries with an implicit neural representation (INR) and passes each candidate through a printability operator before evaluating its final response. The printability operator executes layer-wise self-support, discrete material selection, and whole-body connectivity, assuring fabrication feasibility. To allow for backpropagation through the discrete non-differentiable operations, we use surrogate gradients for the backward pass and a self-supervised printability objective that progressively drives the INR toward a valid design domain. We demonstrate AcoustiPrint for personalized transcranial focused ultrasound, where inter-patient differences in acoustic properties and head geometry deform wave propagation and requires patient-specific acoustic focusing. AcoustiPrint jointly optimizes three-dimensional geometry and material composition through differentiable wave physics while ensuring manufacturability. Experiments show manufacturable acoustic structures with enhanced focal accuracy and over density-based and phase-based baselines, demonstrating a generalized methodology for neural design through non-differentiable manufacturing constraints and differentiable physics.

Introduction

AcoustiPrint addresses the mismatch between physics-optimized and fabricated geometries by placing a printer-parameterized feasibility operator in neural inverse design. It applies this approach to patient-specific ultrasound lens design, with improved focal accuracy and acoustic energy concentration over phase- and density-based baselines.

  • Existing inverse-design methods restrict design parameterizations, use soft manufacturability penalties, or post-process designs, risking a mismatch between simulated and fabricated geometry.Source: Introduction · p. 1
  • Across patients, cortical-skull sound speed varies about 30%, and a lens may produce focal-point displacements of 7.67–21.99 mm.The pipeline uses patient CT-derived acoustic models to jointly optimize lens geometry and material composition through differentiable wave physics.Source: Introduction · p. 2
  • AcoustiPrint produces directly printable volumetric lenses with improved focal accuracy and acoustic energy concentration over phase- and density-based baselines.Source: Introduction · p. 2
  • Its feasibility operator applies discrete material selection, layer-wise self-support, and structural connectivity before evaluating designs with physics.Surrogate gradients support learning through discrete operations, while a self-supervised printability objective drives the neural representation toward feasible designs.Source: Introduction · pp. 1–2

Related Works

Related work addresses differentiable physics, neural design for additive manufacturing, and acoustic lens design, but leaves a gap in jointly optimizing acoustic performance and manufacturability.

  • Differentiable inverse design optimizes physical objectives but often assumes optimized geometries can be manufactured exactly.Source: Related Works · p. 2
  • Neural additive-manufacturing methods use generative design, self-support penalties, or differentiable slicing, but primarily target structural performance [Nie et al. 2021; Chen et al. 2024; Liu et al. 2024, 2025b].Source: Related Works · p. 2
  • Phase-delay lenses use thickness-based, single-material designs, while density optimization does not explicitly enforce manufacturability [Hwan Lee et al. 2025; Stanziola et al. 2023b; Hyun and Kim 2021].AcoustiPrint instead jointly optimizes volumetric acoustic performance and manufacturability.Source: Related Works · p. 2

Method

AcoustiPrint optimizes an implicit acoustic geometry through a printer-parameterized feasibility operator, so the differentiable physics solver evaluates printable designs. Surrogate gradients and a printability objective make the discrete constraints trainable.

Problem Formulation

  • The printer-parameterized operator maps candidate designs into the feasible set before physics evaluation, keeping the simulator’s input printable at every iteration.The formulation treats manufacturability as an operator on the design rather than a penalty competing with the physical objective.Source: Method · p. 2, Problem Formulation · p. 3

Neural Design Parametrization

  • A SIREN implicit neural representation emits occupancy values that set spatial material properties, while the focusing loss maximizes RMS pressure within the target region.The unconstrained reference optimizes the continuous field without fabrication constraints and serves as an empirical upper bound on focusing performance.Source: Neural Design Parametrization · pp. 3–4

Differentiable Printability Operators

  • The printability operator combines discrete material selection, layer-wise slicing and self-support, and extraction of the component grounded at the build plate.Its formulation covers layered additive processes with a fixed build direction, not non-planar or multi-axis deposition.Source: Differentiable Printability Operators · p. 4, Non-Differentiable Printability… · pp. 4–5
  • Support is propagated recursively through each layer, so a voxel’s printability depends on the material beneath it rather than an independent local overhang test.The recursion also propagates gradients down the stack, signaling how a layer change affects buildability above it.Source: Differentiable Printability Operators · p. 4

Surrogate Backpropagation

  • The exact printable geometry is used in the forward pass, while a structurally matched differentiable surrogate supplies gradients that reflect each voxel’s structural role.The surrogate approximates global grounding with local differentiable operations because connected-component labeling has no natural relaxation.Source: Surrogate Backpropagation · p. 5
  • A self-supervised loss penalizes disagreement between the raw INR field and its printable counterpart, progressively reducing the material removed by projection during training.The printable lens directly supervises the continuous field from which it was derived.Source: Surrogate Backpropagation · p. 5

Training Objective

  • The total objective combines focusing and printability losses; its weight controls how strongly the continuous field is pulled toward feasibility, not whether the projected design is printable.The feasibility operator guarantees printability regardless of the weight.Source: Training Objective · p. 5

Evaluation

Evaluation compares AcoustiPrint with phase-only and density-based baselines and examines printability, focusing conditions, and simulator agreement. The reported results cover center and noncentral targets, pattern matching, and slicer effects.

Evaluation

  • Evaluation uses 290 Qure.ai head CT scans and reports five focal-region quantities, including overlap, concentration, shift, peak pressure, and RMS pressure.Unless otherwise stated, simulations use a 10 × 10 grid of transducers at 500 kHz, with the lens covering 60% of the head surface.Source: Evaluation · pp. 5–6

Results

  • For center focusing, the unconstrained reference and AcoustiPrint outperform baselines on all five metrics.The reported results average CT-105, CT-166, and CT-260 for a sphere of radius 4 mm.Source: Table 1 · p. 6, Results · p. 6

Acoustic Lens Comparisons

  • After slicing, the printable lens’s pressure field retains 0.998 cosine similarity, while the unconstrained lens’s field falls to 0.83.The printable lens geometry itself has 0.99 cosine similarity before versus after slicing; the unconstrained lens has 0.84.Source: Figure 2 · p. 6, Acoustic Lens Comparisons · p. 6

Printability Ablation Study

  • Among printable lenses, Surrogate has the best RMS pressure focus and learns structural metrics comparable to the Printable pipeline.These include fewer supported components and removable voxels, and a larger connected component.Source: Printability Ablation Study · p. 6, Figure 3 · p. 7

Other Experiments

  • The 250 kHz-trained lens performs best despite skull attenuation; focusing becomes more complex as targets move away from the head’s geometric center.AcoustiPrint also focuses on letter patterns placed on a 5 mm-thick YZ slice, with extension to XY and XZ slices described.Source: Table 2 · p. 7, Other Experiments · p. 7, Table 3 · p. 7

Cross Simulator Validation

Conclusion

AcoustiPrint designs manufacturable acoustic geometries with a differentiable neural framework and slicer. The paper reports superior performance to four baselines and demonstrates transcranial acoustic focusing.

  • AcoustiPrint and its unconstrained counterpart demonstrate superior performance to four baselines.Source: Conclusion · p. 7
  • A differentiable slicer emulates 3D slicers and teaches the implicit geometry to enforce printability.Source: Conclusion · p. 7
  • The framework is applied to transcranial acoustic focusing, with core ideas described as extendable to other applications.Source: Conclusion · p. 7

Analysis of Inter-Patient Variability

The analysis documents substantial inter-patient variation in acoustic properties and head structure, and reports misfocusing when lenses are transferred between patients.

  • Across 290 CT scans, skull speed and density vary substantially, while brain soft tissue has standard deviations of 7.10 m/s in speed and 5.4 kg/m3 in density.The reported lack of brain-tissue heterogeneity is attributed to CT scans’ limited ability to capture it.Source: Analysis of Inter-Patient Variability · p. 10
  • Tissue composition, skull surface shape, and skull volume vary across patients; the analysis compares surfaces with chamfer distance and volumes with intersection-over-union.Source: Analysis of Inter-Patient Variability · p. 10, Figure 7 · p. 10
  • The transfer analysis selects CT pairs from four acoustic-versus-structural similarity quadrants, fits a lens on one scan, and evaluates its misfocus on the paired scan after ICP alignment.Source: Analysis of Inter-Patient Variability · pp. 10–11
  • The acoustically and structurally similar CT-105/CT-11 pair has the least central-focus misfocus, while dissimilar CT-260/CT-318 has the largest misfocus for central and off-center focusing.Source: Analysis of Inter-Patient Variability · p. 11
  • 29.593 mm of misfocusing separates the thalamus and cerebral cortex in the reported transfer analysis, underscoring the importance of personalization for millimeter-scale focusing.Source: Analysis of Inter-Patient Variability · p. 11

CT-to-Acoustic Conversion

CT Hounsfield units are converted to acoustic properties using density calibration, a linear sound-speed mapping, and tissue-class attenuation assignments.

  • Density is recovered from Hounsfield units using k-Wave’s hounsfield2density calibration.Source: CT-to-Acoustic Conversion · p. 11
  • Sound speed is mapped linearly from density as c = 1.33ρ + 167, with ρ in kg/m3 and c in m/s.Source: CT-to-Acoustic Conversion · p. 11
  • Because no agreed-upon HU-to-attenuation conversion exists, the method assigns constant attenuation by tissue class: 13.3 dB/cm/MHz for skull, 0.55 for brain soft tissue, and 0.02 for cerebrospinal fluid.Source: CT-to-Acoustic Conversion · p. 11

Domain Configuration

The simulation domain is configured around the whole head, with consistent resampling and a prescribed source-and-lens setup for the main experiments.

  • The domain is cropped to fit the head so optimization accounts for whole-head physics, with a 10-grid-point perfectly matched layer absorbing outgoing waves.Source: Domain Configuration · p. 11
  • Every domain is resampled to 0.4 mm for consistent resolution across patients and well-defined lens transfer.Source: Domain Configuration · p. 11
  • Radial transducers have Gaussian apertures and are embedded in the lens; element radius is maximized subject to non-overlap at the prescribed head coverage.Source: Domain Configuration · p. 12
  • Main experiments use a 10×10 transducer grid at 500 kHz, embedded in the lens’s first layer, with 60% head coverage and central or off-center 4 mm-radius targets.Source: Domain Configuration · p. 12

Homogeneous Phased-Array Transducer

The section describes phase-based transducer designs and lens baselines for acoustic focusing. It contrasts homogeneous-medium phase calculations with patient-specific propagation and thickness-based phase compensation.

Homogeneous Phased-Array Transducer

  • Homogeneous PAT sets each transducer element’s phase from its distance to the focus, referenced to the nearest element, while keeping amplitudes at unity.The homogeneous calculation assumes a uniform water medium.Source: Homogeneous Phased-Array Transducer · p. 12
  • Heterogeneous PAT solves the frequency-domain Helmholtz equation with each element active and uses the focus-volume response to compensate phase aberrations by phase conjugation.Only phase is corrected; amplitude remains fixed to unity.Source: Homogeneous Phased-Array Transducer · p. 12
  • Thickness-based lenses convert required phase delays into a thickness profile, wrap thickness to one 2π cycle, and discretize the map onto the simulation grid.The thickness is constrained to the interval from zero to the maximum allowable thickness.Source: Homogeneous Phased-Array Transducer · p. 12
  • The sigmoid density baseline interpolates material properties continuously and then applies a hard threshold at τ = 0.5 to make occupancy binary, without a fabrication constraint.The baseline uses logistic-sigmoid occupancy and water properties to define the interpolation.Source: Homogeneous Phased-Array Transducer · p. 12

Supplementary Visualizations

The supplementary material shows pressure landscapes and focus locations, then examines SIREN and printability settings alongside off-center focusing, transducer-grid, and lens-coverage results.

Supplementary Visualizations

  • The supplementary pressure landscapes show XY, YZ, and XZ slices at 250 and 500 kHz, while the focus-location figure shows deep-center, shallow-center, and lateral targets.The lateral targets are right lateral and left lateral.Source: Figure 9 · p. 12, Figure 10 · p. 13

SIREN parameter ablation

  • Increasing SIREN ω from 30 to 90 raises overlap from 22.9% to 28.0%; ω = 90 is selected for its trade-off between focal accuracy and pressure concentration.At ω = 120, overlap does not improve, both peak and RMS pressure fall, and focal shift rises slightly; this ablation uses CT-120 rather than the main-results CT-105.Source: SIREN parameter ablation · pp. 12–13, Table 7 · p. 13
  • Gumbel temperature τ = 0.05 achieves the highest ratio (1.817) and substantially higher peak and RMS pressure, while lower temperatures achieve higher overlap.The study selects τ = 0.05 because accurate, high-concentration focusing is the primary objective.Source: SIREN parameter ablation · p. 13, Table 8 · p. 13
  • The sigmoid gate temperature τg = 0.01 is selected as superior across all five metrics, although increasing τg generally slightly reduces overlap while marginally improving ratio and pressure metrics.Smaller gate temperatures create a steeper transition toward binary occupancy.Source: SIREN parameter ablation · p. 13, Table 9 · p. 13
  • The printability-loss ablation reports that λ = 0.01 gives peak pressure and lowest shift, but λ = 0.02 is used in subsequent experiments.λ balances emphasis on printability against the acoustic objective.Source: SIREN parameter ablation · p. 13, Table 10 · p. 13

Supplementary Results

  • For off-center focusing, heterogeneous PAT has smaller focal shift than unconstrained and AcoustiCraft designs, but lower overlap and pressure metrics because energy spreads over a broader region.The text attributes its smaller shift to compensation for spatial variation in skull acoustic properties.Source: Supplementary Results · p. 13, Table 11 · p. 13
  • Increasing the number of transducers decreases volume overlap and focal shift, while peak and RMS pressure plateau as transducer size decreases.The lens covers a constant 60% of the head, with individual transducer size set by the grid.Source: Supplementary Results · pp. 13–14, Table 12 · p. 14
  • Larger lens coverage increases acoustic focal accuracy and concentration, giving the INR more freedom to create a better-performing lens.The supplied table is identified as results for varying transducer head-coverage percentages.Source: Supplementary Results · p. 14, Table 13 · p. 14
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