Physics-Aware Spiking Neural Networks

• PAS-Net is a family of spiking neural networks that combine event-driven sparse computation with explicit physical structure from PDE residuals or biomechanical constraints.
• The design blueprint includes both conversion-based methods from trained physics-informed models and directly trained architectures using specialized spiking neuron dynamics and surrogate gradients.
• By preserving task-relevant physical consistency while reducing always-on computation, PAS-Net achieves energy efficiency and robust performance in applications such as PDE solving and wearable human activity recognition.

Searching arXiv for PAS-Net and closely related physics-aware spiking neural network papers. Physics-Aware Spiking Neural Network (PAS-Net) denotes a class of spiking architectures that combine event-driven sparse computation with explicit physical structure. In the current literature, the term is used both as a design blueprint for converting trained physics-informed models into spiking form and as the name of directly trained spiking systems whose physical awareness arises from PDE residuals, separable operator structure, or biomechanical constraints. Across these usages, the unifying objective is to preserve task-relevant physical consistency while reducing dense always-on computation through spike-gated processing (Zhang et al., 2023, Garg et al., 8 Nov 2025, Garg et al., 23 Mar 2026, Zheng et al., 12 Apr 2026).

1. Terminological scope and representative forms

PAS-Net is not restricted to a single network topology. The literature contains at least four representative realizations: a converted PAS-Net obtained from a trained PINN or SPINN, a spiking PINN variant implemented as NeuroPINNs, a separable spiking operator learner implemented as SPINONet, and a fully multiplier-free wearable HAR architecture explicitly named PAS-Net.

Form	Physics-aware mechanism	Spiking mechanism
Converted PINN/SPINN PAS-Net	PDE residuals are learned in the ANN/PINN and transferred after conversion	Offline ANN/PINN $\rightarrow$ SNN conversion with calibration
NeuroPINNs as PAS-Net	PINN residual loss with stochastic-projection derivatives	Variable Spiking Neurons with surrogate backpropagation
SPINONet as PAS-Net	Coordinate derivatives confined to a continuous separable trunk	Event-driven spiking branch for operator coefficients
Wearable PAS-Net	Adaptive symmetric topology mixer enforces human-joint physical constraints	Dynamic-threshold spiking blocks with early exit

The term “physics-aware” is therefore used in two related senses. In scientific machine learning, it usually refers to residual-based enforcement of governing equations or to architectural separation that preserves coordinate differentiability. In wearable HAR, it refers to spatial priors derived from biomechanical topology, specifically human-joint reciprocity encoded by a symmetric routing matrix. This dual usage is explicit in the cited works rather than incidental (Zhang et al., 2023, Garg et al., 23 Mar 2026, Zheng et al., 12 Apr 2026).

2. Conversion-based PAS-Net from PINNs and SPINNs

In the conversion-based formulation, the starting point is a trained ANN or PINN with layerwise outputs

$$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$

which is converted into a rate-coded SNN whose time-averaged output over a window $T$ is

$\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$

The stated objective is to make the final averaged spiking output $\bar{s}^{(n)}=\mathcal{S}(X)$ close to the ANN output $x^{(n)}=\mathcal{A}(X)$.

The conversion error for layer $l$ is decomposed as

$e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$

This formulation separates propagation error from local approximation error and is central to the argument that physics learned in the PINN can be transferred to the SNN without retraining the SNN on physics losses.

The calibration procedure extends classic ANN-to-SNN conversion beyond ReLU to arbitrary activations. For each layer, calibration may be “light,” adjusting only the bias term, or “advanced,” adjusting both scale and shift and optionally small corrections to the weights under a trust region. The layerwise approximation problem is posed as

$\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$

For non-ReLU activations, negative thresholds are introduced to represent positive and negative spikes, which is specifically motivated for odd or bounded activations such as $\tanh$.

The theoretical guarantee assumes that $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$0 and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$1 can be uniformly approximated by piecewise linear functions and that $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$2 is bounded. Under those assumptions, the final-output conversion error is bounded by a weighted sum of local conversion errors: $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$3 The paper’s interpretation is that a PINN’s PDE residual satisfaction is retained up to controlled approximation error after conversion. Conversion and calibration are performed offline after ANN/PINN training, no surrogate gradients are required, and no additional physics-loss fine-tuning is performed on the SNN. In this sense, the converted PAS-Net is physics-aware because it inherits the trained PINN’s residual structure rather than learning it natively in the spike domain (Zhang et al., 2023).

3. Directly trained PDE-solving PAS-Nets: NeuroPINNs

NeuroPINNs implement a PAS-Net as a spiking physics-informed solution network trained directly with PINN losses. The core neuron is the Variable Spiking Neuron (VSN), which replaces the first two hidden-layer activations while leaving the final layers continuous to stabilize regression output. In discrete spike time steps, the VSN dynamics are

$$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$4

with hard reset when $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$5, and graded output

$$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$6

This is explicitly not presented as a continuous-time LIF or GLIF model; it is a discrete-time leak–accumulate–threshold–reset process with graded spikes.

The physics-informed objective is the standard PINN decomposition

$$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$7

with PDE, boundary, and initial-condition terms evaluated at collocation points. The central methodological issue is that direct autograd through the Heaviside nonlinearity is ill-defined, while smoothing the forward model would bias the PDE residual. NeuroPINNs therefore compute spatial and temporal derivatives entering the residual by a stochastic projection (SP) estimator rather than by differentiating through spikes.

At a collocation point $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$8, with neighborhood points $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$9, the SP gradient estimator is

$T$0

Higher-order derivatives are obtained recursively by reapplying the operator. The paper’s stated reason for this construction is to avoid the residual bias that would arise if one replaced $T$1 with a smooth surrogate inside the residual itself.

Surrogate gradients remain present, but only in backpropagation for parameter updates. The forward pass uses true spiking VSN dynamics; the residual uses SP-based derivatives; the backward pass replaces the spike-gate derivative by a bounded surrogate. This separation between physics evaluation and gradient propagation is the distinctive reconciliation mechanism of NeuroPINNs. The same framework is extended to 3D linear elastic micromechanics, where a 3D ResNet with VSN activations is coupled to a non-trainable physics head computing strain, stress, and equilibrium residuals (Garg et al., 8 Nov 2025).

4. Operator-learning PAS-Nets: SPINONet

SPINONet instantiates PAS-Net at the operator level rather than as a pointwise solution network. Its declared objective is to learn an operator mapping that generalizes across input functions and parameters while preserving the coordinate differentiability required for PDE residuals. The design principle is architectural separation: spiking is confined to the input-function encoding pathway, whereas the coordinate pathway remains continuous.

The architecture follows a separable DeepONet-style decomposition with two pathways. The branch is spiking, regression-friendly, and event-driven, producing latent coefficients $T$2 by Variable Spiking Neurons. The trunk is continuous and coordinate-differentiable, with one-dimensional trunk networks $T$3 for each coordinate. The separable basis is assembled as

$T$4

and the operator output is

$T$5

Because coordinate derivatives act only on the trunk,

$T$6

so physics residuals can be computed by automatic differentiation along a fully continuous path.

This separation is the central physics-compatibility argument. Spiking discontinuities are excluded from the coordinate-differentiated route, which prevents contamination of residual derivatives. The branch still benefits from event-driven sparsity. Its VSN dynamics are

$T$7

with surrogate derivative

$T$8

In operator learning, $T$9 is often used to avoid unrolling overhead while preserving sparse event-driven computation.

SPINONet’s efficiency claims are tied not only to spiking activity but also to separability. A non-separable trunk requires evaluations proportional to $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$0 over a $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$1-dimensional grid, whereas the separable trunk reduces this to $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$2. The paper further argues that forward-mode AD is particularly favorable because each trunk takes a scalar input. A hybrid objective with limited data supervision can be added when purely physics-informed training converges to spurious solutions, as in the Eikonal case with global sign flips (Garg et al., 23 Mar 2026).

5. PAS-Net for IMU-based human activity recognition

The architecture explicitly titled PAS-Net in the HAR literature addresses continuous IMU-based human activity recognition under stringent energy and buffering constraints. Here, “physics-aware” refers to biomechanical structure rather than PDE residuals. The network ingests a continuous tensor $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$3, where $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$4 is window length, $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$5 denotes IMU channels, and $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$6 is the number of physical sensor nodes. A rotation-invariant tokenizer extracts patch-wise features with stride $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$7, reducing the temporal dimension to $\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$8. Temporal-aware batch normalization pools statistics over time and nodes,

$\bar{s}^{(l)} = \mathrm{IF}\!\big(W^{(l)} \bar{s}^{(l-1)}\big) \approx \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big), \qquad \bar{s}^{(0)}=x.$9

after which initial LIF neurons emit spikes and the deep core becomes spike-driven.

The central spatial component is the adaptive symmetric topology mixer. For learnable adjacency $\bar{s}^{(n)}=\mathcal{S}(X)$0, strict bidirectional symmetry is enforced by

$\bar{s}^{(n)}=\mathcal{S}(X)$1

Masked causal spatiotemporal mixing is then performed by

$\bar{s}^{(n)}=\mathcal{S}(X)$2

with empirical kernel size $\bar{s}^{(n)}=\mathcal{S}(X)$3, followed by residual clamping to preserve binary outputs. Because the spike tensor is binary, multiplication by spikes is reduced to predicated addition, and the deep spike core is implemented using sparse integer accumulations and conditional moves.

The temporal component is an $\bar{s}^{(n)}=\mathcal{S}(X)$4-memory causal neuromodulator. The gate state is updated as an exponential moving average,

$\bar{s}^{(n)}=\mathcal{S}(X)$5

and modulates the dynamic threshold of a true Dynamic LIF neuron: $\bar{s}^{(n)}=\mathcal{S}(X)$6

$\bar{s}^{(n)}=\mathcal{S}(X)$7

High gate values lower the threshold toward $\bar{s}^{(n)}=\mathcal{S}(X)$8 during bursts, whereas low gate values raise it toward $\bar{s}^{(n)}=\mathcal{S}(X)$9 during idle phases.

PAS-Net’s training objective is the Temporal Spike Error loss,

$x^{(n)}=\mathcal{A}(X)$0

which provides dense supervision over time and enables confidence-driven early exit for streaming inference. The recommended runtime policy uses softmax confidence and margin thresholds, with representative settings $x^{(n)}=\mathcal{A}(X)$1–$x^{(n)}=\mathcal{A}(X)$2, $x^{(n)}=\mathcal{A}(X)$3, and $x^{(n)}=\mathcal{A}(X)$4–$x^{(n)}=\mathcal{A}(X)$5 consecutive steps. The energy model distinguishes FP32 MACs in the shallow stem from sparse AC-only operations in the deep core: $x^{(n)}=\mathcal{A}(X)$6 This PAS-Net is therefore “physics-aware” through symmetric topology masking and context-aware spiking thresholds rather than through explicit PDE residuals (Zheng et al., 12 Apr 2026).

6. Empirical profile, limitations, and recurring misunderstandings

In scientific machine learning, conversion-based PAS-Net demonstrates that calibrated SNNs can retain the performance envelope of trained PINNs across multiple PDEs. Reported errors include Poisson with $x^{(n)}=\mathcal{A}(X)$7 and relative $x^{(n)}=\mathcal{A}(X)$8, diffusion–reaction with $x^{(n)}=\mathcal{A}(X)$9 and relative $l$0, wave with $l$1 and relative $l$2, Burgers with $l$3 and relative $l$4, Burgers with SPINN at $l$5 and relative $l$6, and Navier–Stokes Beltrami flow with SPINN at $l$7 and relative $l$8. Corresponding spike-rate statistics range from $l$9 to $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$0. Error decreases approximately like $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$1 for $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$2 and then saturates; calibration is described as crucial because uncalibrated SNNs lose sharp features and misplace steep gradients. Runtime gains are reported for SPINN-SNN conversion: approximately $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$3 faster than PINN training on 2D Burgers and more than $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$4 faster on 3D Beltrami flow. Hardware energy measurements are not reported, so the energy argument is based on low spike rates and expected neuromorphic efficiency rather than device-level validation (Zhang et al., 2023).

For directly trained PDE PAS-Nets, NeuroPINNs report relative $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$5 errors of $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$6, $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$7, $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$8, and $e^{(l)} \equiv x^{(l)} - \bar{s}^{(l)} = \underbrace{f\!\big(W^{(l)} x^{(l-1)}\big) - f\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{rate/input mismatch } e_r^{(l)}} + \underbrace{f\!\big(W^{(l)} \bar{s}^{(l-1)}\big) - \ClipFloor\!\big(W^{(l)} \bar{s}^{(l-1)}\big)}_{\text{local conversion error } e_c^{(l)}}.$9 on four benchmark PDEs, compared with SP-PINN values of $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$0, $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$1, $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$2, and $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$3; with two spike time steps, the errors improve to $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$4, $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$5, $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$6, and $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$7. SB-NeuroPINN, which uses surrogates inside residual computation, records $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$8, $\min_{\alpha^{(l)},\beta^{(l)},\,V_{\mathrm{th},\pm}^{(l)}} \mathbb{E}\!\left[\big\|\,f\!\big(z^{(l)}\big) - \ClipFloor\!\big(\alpha^{(l)} z^{(l)} + \beta^{(l)}\big) \big\|_2^2\right].$9, DNC, and $\tanh$0, while CPINN gives $\tanh$1, $\tanh$2, DNC, and $\tanh$3. Measured activity yields energy ratios $\tanh$4, $\tanh$5, $\tanh$6, and $\tanh$7, corresponding to synaptic-operation energy savings of approximately $\tanh$8, $\tanh$9, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$00, and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$01 relative to always-on dense ANN layers. In 3D linear elastic micromechanics, the mean relative error in von Mises stress is $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$02 for the spiking variant versus $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$03 for the vanilla non-spiking model; all but $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$04 of $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$05 VSN layers have activity below $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$06 (Garg et al., 8 Nov 2025).

SPINONet’s empirical position is that operator-level PAS-Nets can remain competitive with dense operator learners while scaling more favorably with dimensionality and grid resolution. Reported relative $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$07 errors are $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$08 for viscous Burgers, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$09 for the parametric heat equation, and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$10 for the Eikonal equation, with average branch spiking activities of approximately $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$11, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$12, and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$13, respectively. The separable baseline gives $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$14, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$15, and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$16, while PI-DeepONet gives $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$17 on Burgers, does not converge on the heat equation because of memory limits, and gives $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$18 on Eikonal. The paper explicitly notes that physics-only training may converge to spurious solutions such as global sign flips and that limited data supervision can stabilize those regimes (Garg et al., 23 Mar 2026).

In wearable HAR, PAS-Net is evaluated on seven public IMU datasets under strict subject-independent $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$19 splits. Reported accuracies include PAMAP2 at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$20 versus ResNet-SE at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$21 and DeepConvLSTM at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$22, USC-HAD at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$23 versus $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$24, TNDA-HAR at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$25 versus Self-HAR at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$26, HuGaDB at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$27, Daily-Sports at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$28, HAR70+ at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$29, and Parkinson FOG at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$30. On PAMAP2, PAS-Net uses $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$31 per inference versus ResNet-SE at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$32 and DeepConvLSTM at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$33; on TNDA-HAR it uses $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$34 versus Self-HAR at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$35. Early exits occur as early as $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$36 for HAR70+, PAMAP2, Parkinson, and HuGaDB, with reported dynamic energy savings up to $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$37, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$38, $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$39, and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$40, whereas USC-HAD exits at $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$41 with only $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$42 savings. The architecture also reports approximately $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$43M parameters, about $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$44M nominal FLOPs in the stem, and about $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$45M SOPs in the core on PAMAP2 (Zheng et al., 12 Apr 2026).

Several distinctions are important for interpreting the PAS-Net literature. First, physics awareness is not uniform across papers: it may mean inherited PINN residual satisfaction after conversion, native residual-based training with unbiased derivative estimation, operator architectures that preserve coordinate differentiability, or biomechanical constraints encoded by symmetric topology routing. Second, multiplier-free execution is not universal: it is a defining property of the HAR PAS-Net deep core, whereas NeuroPINNs use graded spikes and the conversion-based and operator-learning variants are primarily framed in terms of sparsity, reduced MACs, or reduced memory traffic rather than strict elimination of multipliers. Third, surrogate gradients are not always the same object: they are unnecessary in the conversion-based PAS-Net, are used only for parameter updates in NeuroPINNs, and are standard training machinery in SPINONet and wearable PAS-Net. Fourth, energy claims are method-dependent: some are based on 45 nm CMOS cost models or analytical operation counts, some include reported dynamic energy per inference, and some remain projections tied to low spike-rate regimes rather than hardware measurements.

The main limitations are also heterogeneous. Conversion-based PAS-Net requires activation functions whose $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$46 and $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$47 admit uniform piecewise-linear approximation and whose $$x^{(l)} = f\!\big(W^{(l)} x^{(l-1)}\big), \qquad x^{(0)}=x,$$48 is bounded on the data distribution; very stiff or highly non-smooth activations may violate these assumptions. NeuroPINNs identify residual bias from forward smoothing and show that post hoc conversion can fail badly on irregular domains. SPINONet notes surrogate-gradient bias, degenerate physics-only solutions, and the difficulty of handling very large parameterized inputs in the spiking branch. Wearable PAS-Net states that energy estimates are based on 45 nm CMOS models and that on-chip deployment, sensor displacement, and continual learning remain open issues. Taken together, these constraints indicate that PAS-Net is best understood as a family of physically structured sparse spiking designs rather than a single settled architecture (Zhang et al., 2023, Garg et al., 8 Nov 2025, Garg et al., 23 Mar 2026, Zheng et al., 12 Apr 2026).