Physics-Informed Neural Network for 2D Acoustic Scattering

Solving the Helmholtz equation across wavenumbers ka = 0.5 to 2π with 2–8% L2 error, under 3% for ka ≥ π

Carl Merrigan — March 2026

Problem: Acoustic Wave Scattering

A plane wave impinges on a rigid circular cylinder. We predict the scattered field — the perturbation caused by the obstacle.

\[\nabla^2 \phi_s + k^2 \phi_s = 0\]

The dimensionless parameter ka (wavenumber × radius) controls difficulty: higher ka = more oscillatory field = finer spatial features.

We train from ka = 0.5 (sub-wavelength) through ka = 2π (2 wavelengths per radius), spanning a 12× range of spatial frequency.

Exact Solution (Validation)

The Jacobi-Anger identity expands the incident wave in cylindrical harmonics:

\[\phi_{\text{inc}} = e^{ikr\cos\theta} = \sum_{n=-\infty}^{\infty} i^n J_n(kr)\,e^{in\theta}\]

The sound-hard Neumann condition \(\partial\phi_{\text{total}}/\partial r\big|_{r=a} = 0\) yields the exact scattered field:

\[\phi_s = -\sum_{n=-\infty}^{\infty} i^n \frac{J_n'(ka)}{H_n^{(1)\prime}(ka)}\, H_n^{(1)}(kr)\, e^{in\theta}\]

Converges for N ≈ ka + 20 terms. This exact solution provides pixel-level validation of the PINN — no numerical reference solver needed.

Scattered-Field Formulation

The network outputs \((u,v) = (\text{Re}(\phi_s),\;\text{Im}(\phi_s))\). The total field is reconstructed as \(\phi_{\text{total}} = e^{ikx} + \phi_s\). Three loss terms enforce physics:

PDE RESIDUAL

\[\nabla^2 \phi_s + k^2 \phi_s = 0\]

NEUMANN BC (r = a)

\[\frac{\partial \phi_s}{\partial n} = -\frac{\partial \phi_{\text{inc}}}{\partial n}\]

BGT2 ABC (r = L)

\[\frac{\partial \phi_s}{\partial r} - ik\phi_s + \frac{\phi_s}{2r} = 0\]

TOTAL LOSS

\[\mathcal{L} = \lambda_{\text{pde}}\,\mathcal{L}_{\text{pde}} + \lambda_{\text{bc}}\,\mathcal{L}_{\text{bc}} + \lambda_{\text{abc}}\,\mathcal{L}_{\text{abc}}\]

Scattered-field formulation lets the network focus on the unknown perturbation, avoiding the need to learn the known incident plane wave.

Network Architecture

(x, y)2D coords

→

Fourier Features σ = k, 128-dim

→

Residual MLP 256 × 4 layers

→

(u, v) Re(φ_s), Im(φ_s)

Random Fourier Features

\[\gamma(\mathbf{x}) = [\sin(B\mathbf{x}),\;\cos(B\mathbf{x})]\]

Overcomes spectral bias — standard MLPs struggle with high-frequency functions in low-dimensional domains [Tancik et al., NeurIPS 2020]

Bandwidth \(\sigma\) set to wavenumber \(k\), encoding the physical length scale directly into the feature space.

Design Choices

Residual blocks: skip connections stabilize deep training
Xavier init: balanced gradient flow at initialization
Tanh activation: smooth, matches solution regularity
Complex output: (u, v) avoids branch cuts in polar form

~296K parameters (FF-PINN) vs ~298K (plain MLP baseline)

Training Strategy & Loss Functions

Collocation Sampling

Two-Phase Optimization

Phase 1 — Adam (10K–50K epochs)
Cosine annealing LR; resampling points every 2K epochs
Phase 2 — L-BFGS (200–300 iters)
Strong Wolfe line search; fixed points for stable Hessian

High-ka: wider network (384–512n), more Fourier features (96–128), 50K Adam epochs.

Loss Components

Component	Definition	λ
\(\mathcal{L}_{\text{pde}}\)	\(\frac{1}{N}\sum\|\nabla^2\phi_s + k^2\phi_s\|^2\)	1.0
\(\mathcal{L}_{\text{bc}}\)	\(\frac{1}{N}\sum\|\partial_n\phi_s + \partial_n\phi_{\text{inc}}\|^2\)	10.0
\(\mathcal{L}_{\text{abc}}\)	\(\frac{1}{N}\sum\|\partial_r\phi_s - ik\phi_s + \phi_s/2r\|^2\)	1.0

Evaluation Metrics

vs analytic Bessel/Hankel series, 200×200 grid.

Metric	Definition
L2 Relative	\(\\|\phi_s^{\text{PINN}} - \phi_s^{\text{exact}}\\|_2 / \\|\phi_s^{\text{exact}}\\|_2\) — primary, normalized
Max Error	\(\max\|\phi_s^{\text{PINN}} - \phi_s^{\text{exact}}\|\) — worst-case
Mean Error	\(\bar{e} = \frac{1}{N}\sum\|\phi_s^{\text{PINN}} - \phi_s^{\text{exact}}\|\) — typical

All errors use complex magnitude \(|\phi| = \sqrt{u^2 + v^2}\).

Ablation: Fourier Features

Parameter-matched comparison — FF-PINN (64 features, 256 hidden, 296K params) vs plain MLP (272 hidden, 298K params). Identical 10K Adam + 200 L-BFGS schedules (shorter than production) to isolate architecture.

ka = π (low frequency)

	FF-PINN	Plain MLP
L2 relative	2.42%	2.42%
Max error	2.97%	2.99%

Identical — spectral bias not a bottleneck at low frequency.

ka = 2π (high frequency)

	FF-PINN	Plain MLP	Ratio
L2 relative	10.1%	11.7%	1.15×
Max error	22.7%	47.3%	2.1×

Max error doubles without Fourier features.

      Key insight: The plain MLP’s L-BFGS phase increased error
      (10.9%→11.7%), while the FF-PINN’s L-BFGS halved it (19.6%→10.1%).
      Fourier features create a smoother loss landscape that second-order optimizers can exploit.
    

Absorbing Boundary Conditions

First-Order ABC

\[\frac{\partial \phi_s}{\partial n} - ik\phi_s = 0\]

Assumes purely outgoing plane waves at the boundary. Works on any shape but ignores wavefront curvature.

BGT2 (Second-Order)

\[\frac{\partial \phi_s}{\partial r} - ik\phi_s + \frac{\phi_s}{2r} = 0\]

Adds a curvature correction \(\phi_s/2r\) that accounts for the \(1/\sqrt{r}\) amplitude decay of cylindrical waves [Bayliss, Gunzburger & Turkel, 1982]

Why Circular Domain?

Symmetry match: circular boundary matches the cylindrical geometry of the scatterer — uniform distance to the obstacle from all points
Well-defined normals: square boundaries have corner singularities where the outward normal is undefined → ABC is ill-conditioned
BGT2 requires curvature: the \(\phi_s/2r\) correction is derived for circular boundaries; it has no square-boundary analogue

        Result: Circle + BGT2 gives the most physically consistent setup
        for this cylindrical scattering problem.
      

Production Results

2.00%

Best L2 error (ka=2π)

4×

ka values validated

ka	L2 Rel Error	Mean Error	Max Error	Network	Epochs	Runtime
ka = 0.5	8.23%	1.90%	4.04%	256 / 4L / 64ff	10K+200	12 min
ka = 1.0	3.57%	1.34%	2.99%	256 / 4L / 64ff	10K+200	13 min
ka = π	2.41%	1.09%	2.96%	256 / 4L / 64ff	10K+200	17 min
ka = 2π	2.00%	0.93%	4.37%	384 / 6L / 96ff	50K+300	259 min

All runs: circle boundary, BGT2 ABC, L = 3.0, trained on 4× NVIDIA RTX 4000 Ada. Mean error < 2% everywhere; max error peaks at 4.4% for ka=2π (localized to the shadow boundary).

L2 relative error is highest at low ka because the scattered field amplitude is small (smaller normalization denominator) and these runs received 4× less compute (10K vs 50K epochs, smaller network). The low-ka errors could likely be reduced with comparable training budgets.

Multi-Scale Scattering Physics

Analytic scattered field Re(φ_s) across four wavenumbers, showing how the scattering pattern evolves from a simple dipole at ka = 0.5 to a complex multi-lobe pattern at ka = 2π.

ka = 0.5: Sub-wavelength — weak, dipolar scattering
ka = 1.0: Transition regime — shadow begins to form
ka = π: Resonance — strong diffraction, clear shadow
ka = 2π: Multi-wavelength — complex interference fringes

The PINN must resolve increasingly fine spatial features as ka grows — motivating Fourier features with σ = k.

PINN vs Analytic: Field Comparisons

ka = π

2.41% L2

2.96% max

ka = 2π

2.00% L2

from 49% — 384n/6L/96ff, 50K Adam + 300 L-BFGS

Spatial Error Distribution

Absolute error |PINN − analytic| across all four production wavenumbers. Error is concentrated near the scatterer surface and in the shadow region behind the cylinder — the regions with the steepest field gradients.

ka = 0.5, 1.0: errors uniformly low, smooth fields are easy for the network
ka = π: shadow boundary shows slightly elevated error where the field transitions sharply
ka = 2π: error peaks near the surface where the oscillation wavelength is smallest — addressed by more Fourier features and longer training (50K epochs)

Training Convergence

All runs show the Adam → L-BFGS transition (sharp loss drop at phase boundary). Higher ka requires progressively more compute.

Scaling Across ka

	ka ≤ π	ka = 2π	ka = 3π
Neurons	256	384	512
Fourier feat.	64	96	128
Adam epochs	10K	50K	50K
Wall time	12–17 min	259 min	10+ hrs
L2 error	2–8%	2.0%	68%

ka=3π: loss drops 5 orders of magnitude but L2 only halves — the architecture cannot resolve ~3 wavelengths per radius. Needs multi-scale decomposition or adaptive methods, not just more compute.

Extension: Honeycomb Acoustic Shield

19-circle honeycomb cluster at ka = 2, trained as a pure PINN with no analytic reference solution.

Loss	Value
PDE	4.92e-6
BC (19)	2.00e-7
ABC	1.41e-7
Total	9.06e-6

Shielding: |φ_total| < 0.003 inside cluster.

Honeycomb: Residual Diagnostics

PDE residual (left panel): finite-difference Laplacian, 200×200 grid. Near-zero in inter-circle gaps. Elevated near boundaries due to steep field gradients — inherent to the physics.

BC residual (right panel): autograd normal derivatives, 100 pts/circle (1,900 total). Neumann condition satisfied to O(10⁻³) across all 19 surfaces.

Conclusion: PINNs as a Multi-Scale Probe

The assignment asks whether a PINN can dynamically resolve physical fields at arbitrary length scales — effectively zooming in to reveal finer spatial structure. These results show that it can, but with a clear tradeoff: resolving shorter wavelengths demands exponentially more compute.

Where PINNs Excel

For low-to-moderate spatial frequency (ka ≤ π), the PINN converges to 2–3% L2 error in 12–17 minutes. At these scales, PINNs offer genuine advantages over mesh-based solvers: continuous field access at any point, no mesh generation, and physics enforced by construction. The network acts as a differentiable, resolution-independent surrogate for the field.

1

Multi-scale accuracy. 2–8% L2 across a 12× range of spatial frequency, single architecture.

2

Physics-aware features. Fourier features with σ=k defeat spectral bias at the source.

3

Symmetry-matched BCs. Circle + BGT2 — physics guides the numerics.

The Compute Wall

As ka grows, training cost scales steeply: the ka=2π run needed 5× more epochs and a 50% wider network (259 min vs 12 min). At ka=3π, a 10+ hour run plateaued at 68% L2 despite loss dropping 5 orders of magnitude — the architecture cannot resolve ~3 wavelengths per radius.

This is not surprising: zooming in to finer features is equivalent to probing higher spatial frequencies, and neural networks face a fundamental resolution–compute tradeoff analogous to the Nyquist limit in signal processing.

4

Patience over engineering. The convergent ka=2π run used default loss weights but 2.5× more epochs — 49%→2%. At the frontier, compute budget is the binding constraint.

Next Steps

Multi-scale / domain decomposition
Curriculum training: low→high ka
Residual-adaptive sampling (RAD)

References

M. Raissi, P. Perdikaris, G.E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378, 686–707 (2019).
M. Tancik, P. Srinivasan, B. Mildenhall et al. Fourier features let networks learn high frequency functions in low dimensional domains. NeurIPS (2020).
A. Bayliss, M. Gunzburger, E. Turkel. Boundary conditions for the numerical solution of elliptic equations in exterior regions. SIAM J. Appl. Math. 42(2), 430–451 (1982).
L. Lu, X. Meng, Z. Mao, G.E. Karniadakis. DeepXDE: A deep learning library for solving differential equations. SIAM Rev. 63(1), 208–228 (2021).
S. Wang, X. Yu, P. Perdikaris. When and why PINNs fail to train: A neural tangent kernel perspective. J. Comput. Phys. 449, 110768 (2022).
P.M. Morse, K.U. Ingard. Theoretical Acoustics. Princeton University Press (1968). Ch. 8: rigid cylinder scattering series solution.

Code: github.com/carlm451/helmholtzscatteringpinn