⚡ electrical engineering

Physics-Informed Direction-of-Arrival Estimation Over Distributed Edge Devices

This paper proposes a physics-informed federated learning framework for Direction-of-Arrival estimation that improves performance and ensures convergence by incorporating steering-vector geometry into the local training objective via a manifold-aware regularizer, thereby overcoming the limitations of treating DoA as a generic classification problem across distributed edge devices.

Original authors: Nathan Tatsuta, Rajeev Sahay

Published 2026-07-07

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CC0 1.0

Original authors: Nathan Tatsuta, Rajeev Sahay

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a group of friends, each standing in a different room with a special microphone array. Their job is to figure out exactly where a sound is coming from (like a bird singing or a car honking). This is called Direction-of-Arrival (DoA) estimation.

Usually, to teach a computer to do this well, you'd need to gather all the audio recordings from every friend into one giant central computer. But that's a privacy nightmare (nobody wants to share their raw recordings) and a logistical headache (sending all that data takes forever).

Federated Learning (FL) is the solution to this. Instead of sending the recordings, each friend trains a local "brain" (a model) on their own data and only sends the lessons learned (the model weights) back to a central server. The server averages these lessons to create a smarter global brain.

The Problem:
Standard Federated Learning treats this like a simple multiple-choice quiz. If the true direction is 5 degrees, and the model guesses 6 degrees, it gets a small penalty. If it guesses 30 degrees, it gets a big penalty. But standard algorithms treat all wrong answers as just "wrong." They don't understand that 6 degrees is physically much closer to 5 degrees than 30 degrees is. They ignore the geometry of how sound waves actually hit the microphones.

When friends have different types of data (some hear sounds from the left, some from the right), this "blind" learning causes their local brains to drift in weird directions, making the final global brain confused and inaccurate.

The Solution: Physics-Informed Learning
The authors, Nathan Tatsuta and Rajeev Sahay, propose a new way to train these brains called FedDoA and DoAProx.

Think of it like this:

Standard Training: A teacher tells a student, "You got the answer wrong. Here is the correct answer. Try again."
Physics-Informed Training: The teacher says, "You got the answer wrong. But look at the map of the room! The sound waves hit the microphones in a specific pattern. Even if you don't know the exact answer, you should know that a guess of 30 degrees is physically impossible if the sound is actually at 5 degrees, because the wave pattern would look totally different. Let's penalize you more for guessing 30 degrees than for guessing 6 degrees."

They do this by adding a "Steering-Vector Regularizer."
In simple terms, this is a rule that forces the model to respect the laws of physics. It measures the distance between the predicted sound wave pattern and the actual sound wave pattern. If the model guesses an angle that creates a wave pattern that doesn't match the physics of the microphone array, it gets a heavy penalty.

Two New Methods:

FedDoA: This is the basic version. It adds the physics rule to the training. It teaches the model that "close guesses" are better than "far guesses" by looking at the wave patterns, not just the numbers.
DoAProx: This is the "supercharged" version. It adds a second rule to keep the local brains from drifting too far away from the group average, especially when the friends have very different data. It's like a safety harness that keeps everyone moving in the same general direction while still respecting the physics.

What They Found:

Faster Learning: Because the models are guided by physics from the very start, they learn much faster than standard methods.
Better Accuracy: Even when the friends have very different data (some only hear left, some only hear right), the physics-informed models stay accurate. The standard models get confused and fail.
The Math: The authors proved mathematically that these methods will eventually settle on a good answer (convergence) and showed that the more microphones you have, the better the physics rule works.

In a Nutshell:
The paper introduces a way to teach AI to locate sounds using a group of devices without sharing private data. Instead of just memorizing answers, the AI is taught to respect the laws of physics (how sound waves behave). This makes the AI smarter, faster, and more reliable, especially when the data is messy or unevenly distributed.

Technical Summary: Physics-Informed Direction-of-Arrival Estimation Over Distributed Edge Devices

Problem Statement

Direction-of-Arrival (DoA) estimation is a fundamental task in array signal processing, traditionally solved using classical methods like MUSIC or GLRT that exploit the geometric structure of the array manifold. While deep learning (DL) has achieved comparable performance, particularly in low-SNR and unknown channel state information (CSI) conditions, it typically requires centralizing large datasets, raising privacy and communication concerns in distributed sensing systems.

Federated Learning (FL) offers a solution by training models locally on edge devices. However, standard FL algorithms (e.g., FedAvg, FedProx, SCAFFOLD) treat DoA as a generic classification problem. They minimize cross-entropy loss where all angular misclassifications are penalized equally. This approach ignores the underlying physics of the array manifold, failing to distinguish between small angular errors (e.g., $5^\circ$ vs. $6^\circ$ ) and large errors (e.g., $5^\circ$ vs. $30^\circ$ ). Under heterogeneous data distributions (non-IID), this mismatch causes client models to drift in directions geometrically inconsistent with the array manifold, degrading the global model's accuracy.

Methodology

The authors propose a physics-informed FL framework that explicitly incorporates the geometry of the Uniform Linear Array (ULA) steering manifold into the local training objective.

Signal Model

The system considers a ULA with $M$ elements. The received signal vector $z(n)$ is modeled as:
$z(n) = a(\theta)s(n) + n(n)$
where $a(\theta)$ is the steering vector encoding the array geometry. A key property utilized is that angular proximity implies steering-vector proximity ( $\|a(\theta_1) - a(\theta_2)\|$ grows with angular separation).

Proposed Frameworks

The authors introduce two specific FL objectives:

FedDoA: This method modifies the standard local loss $f_k(w)$ by adding a steering-vector regularizer:
$L^D_k(w) = f_k(w) + \frac{\beta}{2} \|a(\hat{\theta}_k) - a(\theta_k)\|^2$
Here, $\hat{\theta}_k$ is the predicted DoA, $\theta_k$ is the true DoA, and $\beta$ is a weighting hyperparameter. Unlike standard approaches that penalize label space discrepancies, this term penalizes Euclidean distance in the steering space, enforcing geometry-consistent penalties.
DoAProx: This method combines the FedDoA steering regularizer with the proximal term from FedProx to further mitigate client drift under severe heterogeneity:
$L^{DP}_k(w) = f_k(w) + \frac{\mu}{2} \|w - w^{(0)}\|^2 + \frac{\beta}{2} \|a(\hat{\theta}_k) - a(\theta_k)\|^2$
The proximal term constrains local weights to remain close to the global model $w^{(0)}$ , while the steering term enforces angular correction on the manifold.

Convergence Analysis

The paper provides rigorous non-convex convergence analysis without assuming Polyak–Łojasiewicz conditions or convexity.

Convergence Rate: Both FedDoA and DoAProx are proven to converge to a stationary point at a rate of $O(1/\sqrt{T})$ , where $T$ is the number of communication rounds.
MSAE Bounds: The authors derive the first direct Mean Squared Angular Error (MSAE) bounds for federated DoA. The bounds show that the error decreases as the regularization weight $\beta$ and the array aperture $M$ increase. Specifically, DoAProx achieves a strictly tighter MSAE bound than FedDoA under heterogeneous data by suppressing client drift.

Experimental Results

The framework was evaluated using a CNN architecture (based on [15]) across 180,000 generated signals with $M=8$ elements and $N=1500$ snapshots. Experiments covered four conditions varying in label and channel heterogeneity (IID vs. non-IID) with $K \in \{6, 12\}$ clients.

Performance under IID/Non-IID Labels:
- In label-IID conditions, physics-informed methods (FedDoA, DoAProx) converged faster (within 50–80 rounds) and reached higher final accuracy (90–95%) compared to baselines (FedAvg, FedProx, SCAFFOLD).
- In label-non-IID conditions (disjoint angular sectors), all methods saw a drop in accuracy (60–70%). However, FedDoA and DoAProx consistently outperformed baselines. The steering-vector regularizer provided a stable gradient signal robust to angular distribution mismatches, guiding local updates toward better global generalization.
Channel Heterogeneity: The channel distribution (Gaussian, Rayleigh, Rician) had no significant effect on the performance gap between physics-informed and non-physics-informed methods, confirming that the gains are driven by the angular prediction geometry rather than channel features.
DoAProx Advantage: In the most challenging scenario (label-non-IID and channel-non-IID with $K=12$ ), DoAProx showed additional benefits by reducing the drift introduced by simultaneous heterogeneities.

Key Contributions and Significance

The paper claims the following specific contributions:

Physics-Informed DoA Estimation: The proposal of the first physics-informed FL framework for DoA that embeds steering-vector geometry directly into the local objective via manifold-aware regularizers (FedDoA and DoAProx).
Convergence Guarantees: The establishment of the first rigorous non-convex convergence analysis for a federated DoA method, proving $O(1/\sqrt{T})$ convergence.
Theoretical Bounds: The derivation of the first direct MSAE bound for federated DoA learning, demonstrating that the bound tightens with both the regularization weight and the array aperture.

Significance: The work demonstrates that incorporating domain knowledge (array manifold physics) into the FL training paradigm significantly improves performance over generic FL baselines, particularly in heterogeneous distributed settings. The approach is network-architecture agnostic and adds negligible computational overhead while providing theoretical guarantees on convergence and angular estimation error.

Future Work: The authors note that future investigations will focus on multi-source generalizations and handling array imperfections such as nonlinear distortions.