Deep Phasor Networks: Phase-Based Neural Models

• Deep Phasor Networks are neural architectures that use unit-magnitude complex numbers (phasors) to encode phase information, enabling efficient modeling of oscillatory data.
• They integrate trainable phase shifts, global DFT-based mixing, and pull-back normalization to maintain geometric constraints and ensure energy preservation.
• Empirical results show these networks achieve competitive accuracy and significant computational efficiency in tasks such as time series forecasting and neuromorphic inference.

Deep Phasor Networks are a class of neural architectures that operate on states encoded as unit-magnitude complex numbers (phasors), with activations parameterized purely by phase on the unit circle $S^1$ or more generally the $N$-torus $\mathbb{T}^N$. These networks exploit phase-native computation, structured global token mixing via unitary operations such as the Discrete Fourier Transform (DFT), and “pull-back” normalization gates to maintain geometric constraints while supporting deep stacking. Their design is motivated by efficiency in sequence modeling, particularly for oscillatory or periodic data, and they subsume architectures such as the Phasor Transformer, Variational Phasor Circuits, and spiking phasor and residual MLP frameworks (Sigdel, 18 Mar 2026, Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026, Olin-Ammentorp et al., 2021, Bybee et al., 2022, Chen et al., 2023, Kazemi et al., 2021).

1. Foundations: Phasor State Representations and Unit Circle Geometry

The core building block in Deep Phasor Networks is the phasor activation $z = e^{i\phi}$, where $\phi$ is a phase variable typically in $(–\pi, \pi]$ or $[0, 2\pi)$. In an $N$-dimensional network, the complete state is $$\mathbf{z} = (e^{i\phi_1}, \ldots, e^{i\phi_N})^\top \in \mathbb{T}^N \subset \mathbb{C}^N$$, constraining all components to the unit torus. This geometric encoding imposes a phase-native inductive bias, making such models inherently well-suited for oscillatory signals and tasks where relative timing or phase plays a fundamental role (Sigdel, 18 Mar 2026, Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026, Olin-Ammentorp et al., 2021).

Phasor activations allow an exact mapping between time-domain spike events and phase, making these models adaptable to temporal and neuromorphic inference: a phase variable can be represented by spike timing within a cycle, and phasor summation by periodic spike trains (Olin-Ammentorp et al., 2021, Bybee et al., 2022).

2. Architectural Components: Gates, Mixing, and Normalization

Deep Phasor Networks assemble their computations from a set of analytic building blocks:

• Trainable Coordinate-Wise Phase-Shifts: Diagonal unitaries $$S(\theta) = \mathrm{diag}(e^{i\theta_1},...,e^{i\theta_N})$$ apply independent, learned phase offsets to each channel. These are the primary trainable parameters per layer in most designs (Sigdel, 18 Mar 2026, Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).
• Unitary Mixing Layers: Global or local mixing is performed using fixed unitary operators. The most salient is the DFT matrix $N$0, defined by $N$1, which entangles all channels and provides $N$2 global token mixing without explicit attention maps (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026). Alternatively, local “beam-splitter” gates or pairwise mixing provided by sparse local unitaries can be stacked to form deep circuits (Sigdel, 18 Mar 2026).
• Pull-Back Normalization: To enforce unit-modulus constraints after each mixing operation and stabilize phase drift, all components are projected back to $N$3 via $N$4 (if $N$5), or phase-folded via functions such as $N$6 (Sigdel, 18 Mar 2026, Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).
• Readout Layers: Output is extracted by inspecting the phase of a designated wire and mapping it to decision probabilities (e.g., $N$7 for binary tasks; softmax of phase magnitudes for multiclass) (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).

A summary table of key operators is as follows:

Operator Type	Mathematical Form	Role
Trainable phase shift	$N$8	Coordinatewise phase gate
Global DFT mixing	$N$9	All-to-all token coupling
Local pair mixing	$\mathbb{T}^N$0	Nearest-neighbor interaction
Pull-back normalization	$\mathbb{T}^N$1	Enforce S$\mathbb{T}^N$2 geometry

3. Deep Stacking: Network Topologies and Scaling

Deep Phasor Networks are realized by stacking layers of shift and mixing gates. For example, the Phasor Transformer block is

$\mathbb{T}^N$3

and the full Large Phasor Model (LPM) applies $\mathbb{T}^N$4 such blocks: $\mathbb{T}^N$5 with parameter count $\mathbb{T}^N$6. Each mixing operation $\mathbb{T}^N$7 is evaluated via FFT, giving subquadratic $\mathbb{T}^N$8 complexity for global mixing, in contrast to the quadratic $\mathbb{T}^N$9 scaling in Transformer attention maps (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).

Variational Phasor Circuits (VPCs) generalize this approach: each block combines a trainable shift, a fixed local or global unitary mixer, and normalization. Stacking $z = e^{i\phi}$0 VPCs yields $z = e^{i\phi}$1 parameters (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026). This linear scaling supports extreme parameter-efficient deep networks.

In residual phasor MLP variants, linear shortcut layers can be added in parallel to deeper nonlinear blocks, for fast convergence and leveraging known physical structure in tasks such as power flow (Chen et al., 2023).

4. Training Protocols, Loss Functions, and Optimization

Training in Deep Phasor Networks leverages phase-aware loss functions and gradient backpropagation through the complex domain or phase manifold:

• Loss functions: Mean squared error (MSE) for regression, cross-entropy for classification, and cosine similarity on phase vectors are adopted. For example, $z = e^{i\phi}$2 for classification targets mapped to specific phases (Olin-Ammentorp et al., 2021, Sigdel, 18 Mar 2026, Sigdel, 18 Mar 2026).
• Gradient computation: Differentiation is computed either via the complex chain rule or Wirtinger calculus, projecting gradients onto the real tangent of $z = e^{i\phi}$3. Automatic differentiation in $z = e^{i\phi}$4 is supported directly in frameworks such as PyTorch (Sigdel et al., 16 Mar 2026, Sigdel, 18 Mar 2026). Pull-back normalization is sub-differentiable almost everywhere, and in practice, gradients are propagated through as identities at unit modulus.
• Optimization: Adam is standard; L-BFGS-B and COBYLA are used in situations where periodicity may induce topologically distinct minima or gradient stalling (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026). Physics-informed initialization dramatically accelerates convergence in hybrid architectures for inverse problems (Chen et al., 2023).

Parameter counts are typically orders of magnitude lower than in classical attention or dense real-valued models: for $z = e^{i\phi}$5, $z = e^{i\phi}$6, the phasor Transformer uses 64 parameters, while comparable attention models reach several thousand (Sigdel, 18 Mar 2026).

5. Empirical Results and Efficiency–Performance Trade-offs

Deep Phasor Networks demonstrate explicit trade-offs between efficiency and predictive performance:

• Time Series Forecasting: On synthetic multi-frequency benchmarks, single-block phasor Transformers achieve MSE $z = e^{i\phi}$7 0.07 (50 params) vs. PyTorch transformer MSE $z = e^{i\phi}$8 0.003 ($z = e^{i\phi}$91,000 params) for context $\phi$0. For longer contexts ($\phi$1), phasor networks approach state-of-the-art at a fraction of the parameter count, validating the value of subquadratic mixing (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).
• Classification: On Brain-Computer Interface (BCI) synthetic EEG signals (N=32), deep VPCs attain 100% binary and 99% four-way accuracy with only 64/128 phase parameters, outperforming dense decision trees and approaching random-forest/SVM/MLP accuracy at much lower parameter cost (Sigdel, 18 Mar 2026).
• Neuromorphic and Spiking Codes: Spiking phasor networks maintain single-spike-per-cycle timing codes, achieving 1.06% MNIST error and 34.09% on CIFAR-10, competitive with real-valued DNNs and superior to other temporal SNNs (Bybee et al., 2022).
• Power Flow Prediction: Deep Phasor Networks trained with physics-guided initialization accelerate AC power flow analysis by 1,000-4,000$\phi$2 compared to Monte Carlo or Newton solvers, achieving near-Newton-accuracy and robust PPF estimation (Chen et al., 2023).
• Phaseless Imaging: Unrolled Wirtinger Flow architectures with deep decoding priors halve the sample complexity in phase retrieval, with MSE improvements of 3–10$\phi$3 over classical algorithms at identical measurement budgets (Kazemi et al., 2021).

Empirical results show that phasor-based architectures are especially advantageous where geometric (phase) structure reflects underlying data (oscillatory, periodic, or interference-dominated domains) and where parameter or resource budgets are a bottleneck.

6. Theoretical Properties, Interpretability, and Extensions

Deep Phasor Networks benefit from several distinctive theoretical and practical advantages:

• Unitary Norm Preservation: All linear circuit stages are unitary; total $\phi$4 energy is preserved, preventing vanishing/exploding activations (Sigdel et al., 16 Mar 2026, Sigdel, 18 Mar 2026).
• Interpretability: Phase interference in readouts (constructive/destructive) and global coherence metrics enable explicit geometric interpretation of model decisions and behavior. Decision boundaries correspond directly to regions of high phase alignment (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).
• Relation to Quantum and Classical Models: Many architectures are inspired by or analogous to variational quantum circuits but operate deterministically and classically in $\phi$5. There are no measurement bottlenecks, no barren plateaus, and easy integration with conventional machine learning frameworks (Sigdel, 18 Mar 2026, Sigdel et al., 16 Mar 2026).
• Hybridization and Extensions: Deep Phasor Networks are applicable as standalone models, as front-ends for hybrid quantum-classical pipelines, as memory systems (via Hebbian phase-coupling), or as anomaly detectors via phase coherence metrics. Extensions under study include photonic/analog implementations, graph-based generalizations, uncertainty-aware layers, and integration with vector-symbolic representations (Sigdel et al., 16 Mar 2026, Chen et al., 2023, Sigdel, 18 Mar 2026).

A plausible implication is that phase-native architectures furnish a scalable, efficient modeling paradigm for long-context sequence processing, oscillatory temporal analysis, and neuromorphic computing, where norm-preservation, interpretable phase boundaries, and lightweight parameterization confer unique advantages. This suggests future survey and benchmarking of Deep Phasor Networks alongside both Euclidean and quantum-inspired alternatives to further clarify domains of supremacy.