docs/research/arena-physica/maxwells-equations-wifi-sensing.md
Research document for wifi-densepose project. Date: 2026-04-02
All WiFi-based sensing ultimately derives from Maxwell's four partial differential equations governing electromagnetic field behavior:
(1) Gauss's Law (Electric): nabla . E = rho / epsilon_0
(2) Gauss's Law (Magnetic): nabla . B = 0
(3) Faraday's Law: nabla x E = -dB/dt
(4) Ampere-Maxwell Law: nabla x B = mu_0 * J + mu_0 * epsilon_0 * dE/dt
In free space with no charges or currents (the indoor propagation case), these simplify to the wave equation:
nabla^2 E - mu_0 * epsilon_0 * d^2 E / dt^2 = 0
yielding plane wave solutions E(r, t) = E_0 * exp(j(k . r - omega * t)) where k = 2*pi / lambda is the wavenumber. At 2.4 GHz WiFi, lambda ~ 12.5 cm; at 5 GHz, lambda ~ 6 cm.
Channel State Information (CSI) is the frequency-domain representation of the wireless channel's impulse response. The derivation from Maxwell's equations proceeds through several simplification layers:
Layer 1: Full Maxwell's equations -- Exact but computationally intractable for room-scale environments at GHz frequencies.
Layer 2: High-frequency ray optics (Geometrical Optics / Uniform Theory of Diffraction) -- When object dimensions >> lambda (walls, furniture), Maxwell's equations reduce to ray tracing. Each ray follows Snell's law at interfaces, with Fresnel reflection/transmission coefficients computed from the dielectric contrast.
Layer 3: Multipath channel model -- The channel impulse response aggregates all propagation paths:
h(t) = sum_{n=1}^{N} alpha_n * exp(-j * phi_n) * delta(t - tau_n)
where for each path n:
alpha_n = complex attenuation (from free-space path loss, reflection, diffraction)phi_n = 2*pi*f*tau_n = phase shifttau_n = d_n / c = propagation delay (distance / speed of light)Layer 4: Channel Frequency Response (CFR) = CSI -- The Fourier transform of h(t):
H(f_k) = sum_{n=1}^{N} alpha_n * exp(-j * 2*pi * f_k * tau_n)
Each OFDM subcarrier k at frequency f_k provides one complex CSI measurement:
H(f_k) = |H(f_k)| * exp(j * angle(H(f_k)))
With 802.11n/ac providing 56-256 subcarriers and 802.11ax up to 512 subcarriers across 160 MHz bandwidth, CSI captures a frequency-sampled version of the channel's multipath structure.
Key insight for sensing: When a human moves in the environment, paths reflecting off the body change their alpha_n, tau_n, and phi_n, modulating the CSI. The sensing problem is to invert this relationship -- recover body state from CSI changes.
The Tsinghua WiFi Sensing Tutorial (tns.thss.tsinghua.edu.cn) identifies two mainstream models:
Ray-Tracing Model: Establishes explicit geometric relationships between signal paths and CSI. The received signal is:
V = sum_{n=1}^{N} |V_n| * exp(-j * phi_n)
This model enables extraction of geometric parameters (distances, reflection points, angles of arrival) from CSI data. It underpins localization and tracking applications.
Scattering Model: Decomposes CSI into static and dynamic contributions:
H(f,t) = sum_{o in Omega_s} H_o(f,t) + sum_{p in Omega_d} H_p(f,t)
Dynamic scatterers (moving bodies) contribute through angular integration:
H_p(f,t) = integral_0^{2pi} integral_0^{pi} h_p(alpha, beta, f, t) * exp(-j*k*v_p*cos(alpha)*t) d_alpha d_beta
The scattering model yields the CSI autocorrelation:
rho_H(f, tau) ~ sinc(k * v * tau)
enabling speed extraction from autocorrelation peak analysis:
v = x_0 * lambda / (2 * pi * tau_0)
where x_0 is the first sinc extremum location and tau_0 is the corresponding time lag.
| Approximation | Physical Basis | Used When | Accuracy |
|---|---|---|---|
| Ray tracing (GO/UTD) | High-frequency limit of Maxwell | Objects >> lambda | Good for LOS + major reflections |
| Fresnel zone model | Wave diffraction | Target near TX-RX line | Excellent for presence/respiration |
| Born approximation | Weak scattering (small perturbation) | Low-contrast objects | Breaks down for human body |
| Rytov approximation | Phase perturbation expansion | Moderate scattering | Better for lossy media |
| Free-space path loss | 1/r^2 power decay | Coarse attenuation models | Adequate for RSSI-based sensing |
Relevance to wifi-densepose: Our field_model.rs implements the eigenstructure approach (Layer 2.5 -- between full ray tracing and statistical models), decomposing the channel covariance via SVD to separate environmental modes from body perturbation. Our tomography.rs implements the voxel-based inverse at Layer 3 using L1-regularized least squares.
Physics-Informed Neural Networks embed physical laws as constraints in the loss function or network architecture. For RF sensing, PINNs encode electromagnetic propagation principles:
Standard PINN loss for RF propagation:
L_total = L_data + lambda_physics * L_physics + lambda_boundary * L_boundary
where:
L_data = (1/N) * sum |H_pred(f_k) - H_meas(f_k)|^2 (CSI measurement fit)
L_physics = (1/M) * sum |nabla^2 E + k^2 * E|^2 (Helmholtz equation residual)
L_boundary = (1/B) * sum |E_pred - E_bc|^2 (boundary conditions)
The Helmholtz equation nabla^2 E + k^2 * n^2(r) * E = 0 (time-harmonic Maxwell) constrains the solution space, where n(r) is the spatially varying refractive index.
PINN + GNN for RF Map Construction (arXiv 2507.22513):
PINN for Wireless Channel Estimation (NeurIPS 2025, OpenReview r3plaU6DvW):
ReVeal: High-Fidelity Radio Propagation (DySPAN 2025):
Physics-Informed Generative Model for Passive RF Sensing (arXiv 2310.04173, Savazzi et al.):
Multi-Modal Foundational Model (arXiv 2602.04016, February 2026):
Generative AI for Wireless Sensing (arXiv 2509.15258, September 2025):
Algorithm: Physics-Informed CSI Sensing Network
Input: CSI tensor H[time, subcarrier, antenna] of shape (T, K, M)
Output: Body state estimate (pose, position, or occupancy)
1. PREPROCESSING (physics-guided):
a. Remove carrier frequency offset (CFO): H_clean = H * exp(-j*2*pi*delta_f*t)
b. Conjugate multiply across antenna pairs to cancel common phase noise
c. Compute CSI-ratio: H_ratio(f,t) = H_dynamic(f,t) / H_static(f,t)
2. PHYSICS ENCODER:
a. Embed Fresnel zone geometry as positional encoding
b. Apply multi-head attention with frequency-aware kernels
c. Enforce causality: attention mask respects propagation delay ordering
3. PHYSICS-CONSTRAINED DECODER:
a. Predict body state x_hat
b. Forward-simulate expected CSI from x_hat using ray-tracing differentiable renderer
c. Compute physics loss: L_phys = ||H_simulated(x_hat) - H_measured||^2
4. TRAINING LOSS:
L = L_pose_supervision + alpha * L_phys + beta * L_temporal_smoothness
Our RuvSense pipeline already implements physics-guided preprocessing (phase alignment, coherence gating, Fresnel zone awareness). The next step would be to:
field_model.rs) as an informed priorThe forward problem: given a known body position/shape and room geometry, predict the CSI.
Forward: body_state -> Maxwell/ray-tracing -> H(f,t) [well-posed]
Inverse: H(f,t) -> ??? -> body_state [ill-posed]
WiFi sensing is fundamentally an inverse scattering problem. A WiFi antenna receives signal as 1D amplitude/phase -- the spatial information of the 3D scene is collapsed to a single CSI complex number per subcarrier per antenna pair. Reconstructing fine-grained spatial information from this compressed observation is severely ill-posed.
Helmholtz equation with scatterer:
nabla^2 E(r) + k^2 * (1 + O(r)) * E(r) = 0
where O(r) = epsilon_r(r) - 1 is the object function (dielectric contrast of the body relative to free space).
Born approximation (first-order): Assumes the field inside the scatterer equals the incident field:
E_scattered(r) ~ k^2 * integral O(r') * E_incident(r') * G(r, r') dr'
where G(r, r') is the free-space Green's function. This is valid when O(r) is small and the object is electrically small. For the human body at 2.4 GHz (epsilon_r ~ 40-60 for muscle tissue), the Born approximation is grossly violated.
Rytov approximation: Expands the complex phase rather than the field:
E_total(r) = E_incident(r) * exp(psi(r))
psi(r) ~ (k^2 / E_incident(r)) * integral O(r') * E_incident(r') * G(r, r') dr'
The Rytov approximation handles larger phase accumulation than Born but still assumes weak scattering. It works better for lossy media where absorption limits multiple scattering.
Extended Phaseless Rytov Approximation (xPRA-LM) (Dubey et al., arXiv 2110.03211):
epsilon_r up to 15+j1.5 (20x wavelength size)epsilon_r = 77+j7 (water/tissue), shape reconstruction still accurateFor high-contrast scatterers like the human body, iterative methods are required:
Distorted Born Iterative Method (DBIM):
Algorithm: DBIM for WiFi Body Imaging
Input: Measured scattered field E_s at receiver locations
Output: Object function O(r) (dielectric map of scene)
1. Initialize: O_0(r) = 0 (empty room)
2. For iteration i = 0, 1, 2, ...:
a. Solve forward problem: compute total field E_i(r) in medium with O_i(r)
b. Compute Green's function G_i(r, r') for medium O_i(r)
c. Linearize: delta_E_s = K_i * delta_O (Frechet derivative)
d. Solve: delta_O = K_i^+ * (E_s_measured - E_s_computed(O_i))
e. Update: O_{i+1} = O_i + delta_O
f. Check convergence: ||E_s_measured - E_s_computed(O_{i+1})|| < epsilon
Challenges for WiFi sensing:
RTI (Wilson & Patwari, 2010) simplifies the inverse scattering problem by:
Forward model:
y = W * x + n
where:
y = [y_1, ..., y_L]^T attenuation measurements (L links)
x = [x_1, ..., x_V]^T voxel occupancy values (V voxels)
W = [w_{l,v}] weight matrix (link-voxel intersection)
n = measurement noise
Weight model (elliptical):
w_{l,v} = { 1 / sqrt(d_l) if d_{l,v}^tx + d_{l,v}^rx < d_l + lambda_w
{ 0 otherwise
where:
d_l = distance between TX_l and RX_l
d_{l,v}^tx = distance from TX_l to voxel v center
d_{l,v}^rx = distance from RX_l to voxel v center
lambda_w = excess path length parameter (typically ~lambda/4)
Inverse solution (Tikhonov-regularized):
x_hat = (W^T W + alpha * C^{-1})^{-1} * W^T * y
where C is the spatial covariance matrix and alpha controls regularization.
Our implementation (tomography.rs) uses ISTA (Iterative Shrinkage-Thresholding Algorithm) with L1 regularization for sparsity:
Algorithm: ISTA for RF Tomography (as in tomography.rs)
Input: Weight matrix W, observations y, lambda (L1 weight)
Output: Sparse voxel densities x
1. Initialize x = 0
2. step_size = 1 / ||W^T * W||_spectral
3. For iter = 1 to max_iterations:
a. gradient = W^T * (W * x - y)
b. x_candidate = x - step_size * gradient
c. x = soft_threshold(x_candidate, lambda * step_size)
where soft_threshold(z, t) = sign(z) * max(|z| - t, 0)
d. residual = ||W * x - y||
e. if residual < tolerance: break
Dubey, Li & Murch (arXiv 2311.09633) reconciled empirical RTI with formal inverse scattering theory:
DensePose From WiFi (Geng, Huang, De la Torre, arXiv 2301.00250, CMU):
RF-Pose (Zhao et al., CVPR 2018, MIT CSAIL):
Person-in-WiFi (Wang et al., ICCV 2019, CMU):
3D WiFi Pose Estimation (arXiv 2204.07878):
HoloCSI (2025-2026):
FDTD discretizes Maxwell's curl equations on a Yee grid and marches forward in time:
Algorithm: FDTD Update (2D TM mode, simplified)
Grid: dx = dy = lambda/20 (minimum 10 cells per wavelength)
Time step: dt = dx / (c * sqrt(2)) [Courant condition]
For each time step n:
1. Update H fields:
H_z^{n+1/2}(i,j) = H_z^{n-1/2}(i,j) + (dt/mu_0) * [
(E_x^n(i,j+1) - E_x^n(i,j)) / dy -
(E_y^n(i+1,j) - E_y^n(i,j)) / dx
]
2. Update E fields:
E_x^{n+1}(i,j) = E_x^n(i,j) + (dt / epsilon(i,j)) * [
(H_z^{n+1/2}(i,j) - H_z^{n+1/2}(i,j-1)) / dy
]
For WiFi at 2.4 GHz:
Key references for WiFi FDTD:
Practical limitations: FDTD is too slow for real-time sensing but valuable for:
MoM converts Maxwell's integral equations into matrix equations by expanding fields in basis functions:
[Z] * [I] = [V]
where:
Z_{mn} = integral integral G(r_m, r_n) * f_m(r) * f_n(r') dS dS'
I_n = unknown current coefficients
V_m = incident field excitation
Application: MoM excels for antenna analysis and is used to model WiFi antenna patterns. Less practical for full room simulation due to O(N^2) memory and O(N^3) solve time.
FEM handles complex geometries and material interfaces more naturally than FDTD:
Weak form of Helmholtz equation:
integral nabla x E_test . (1/mu_r * nabla x E) dV - k_0^2 * integral E_test . epsilon_r * E dV
= -j * omega * integral E_test . J_s dV
Application: HFSS (Ansys) and COMSOL use FEM for electromagnetic simulation. Arena Physica's Heaviside-0 model was trained against such commercial FEM solvers.
| Method | Speed | Accuracy | Body Modeling | Room Scale | Real-Time |
|---|---|---|---|---|---|
| FDTD | Hours | Full-wave exact | Excellent | Feasible (GPU) | No |
| MoM | Hours | Exact for surfaces | Good (surface) | Impractical | No |
| FEM | Hours | Exact | Excellent | Feasible | No |
| Ray tracing | Seconds | GO/UTD approximation | Coarse | Easy | Near real-time |
| RTI (ISTA) | Milliseconds | Linear approximation | Voxelized | Easy | Yes |
| Neural surrogate | Milliseconds | Trained accuracy | Implicit | Trained domain | Yes |
The most promising direction combines full-wave accuracy with real-time speed:
This is exactly Arena Physica's approach (Section 5), applied to RF component design rather than sensing. The same methodology applies to WiFi sensing: train a neural forward model on FDTD data, then use it as a differentiable physics constraint during inverse model training.
Arena Physica (arena-ai.com / arenaphysica.com) pursues "Electromagnetic Superintelligence" -- building foundation models that develop superhuman intuition for how geometry shapes electromagnetic fields. Founded by Pratap Ranade (CEO), Arya Hezarkhani, Claire Pan, Michael Frei, and Harish Krishnaswamy. Offices in NYC (HQ), SF, LA.
Raised $30M Series B (April 2025). Deployed with AMD, Anduril Industries, Sivers Semiconductors, Bausch & Lomb. Claims 35% reduction in engineering man-hours and multi-month acceleration in time-to-market.
Arena's Atlas platform uses two foundation models:
Heaviside-0 (Forward Model):
Marconi-0 (Inverse Model):
Training data: 3 million simulated designs across 25 expert templates + random structures, totaling 20+ years of combined simulation time. Incorporates both S-parameter data and electromagnetic field distributions.
Validation: Predictions validated against commercial numerical field solvers (likely HFSS). Internal testing shows < 1 dB magnitude-weighted MAE (RF engineers operate in 20-30 dB ranges).
Arena does NOT solve Maxwell's equations directly. Instead:
As Pratap Ranade states: the model "learns the syntax of physics" inductively from examples, rather than deductively from equations. This trades precision for speed -- acceptable when searching design space where "speed and direction matter more than precision."
Arena's LFM is distinct from Large Language Models:
Arena Physica focuses on RF component design (antennas, PCBs, filters), not WiFi sensing. However, their approach is directly transferable:
| Arena Physica (Design) | WiFi Sensing (Our Case) |
|---|---|
| Forward: geometry -> S-parameters | Forward: body pose -> CSI |
| Inverse: S-parameters -> geometry | Inverse: CSI -> body pose |
| Train on FDTD/FEM simulations | Train on ray-tracing / FDTD simulations |
| 13ms inference | Real-time CSI inference |
| Conditional diffusion for generation | Conditional generation for pose prediction |
Key lesson for wifi-densepose: Building a neural forward model (body_pose -> expected_CSI) trained on electromagnetic simulation data, then using it as a differentiable physics constraint during inverse model training, could significantly improve our pose estimation accuracy and generalization. This is the "physics-informed" approach with the computational burden shifted to offline training.
| Module | Physical Model | Maxwell Connection |
|---|---|---|
field_model.rs | SVD eigenstructure decomposition | Eigenmode basis of room's EM field |
tomography.rs | L1-regularized RTI (ISTA solver) | Linearized inverse scattering |
multistatic.rs | Attention-weighted cross-node fusion | Exploits geometric diversity of multiple TX/RX |
phase_align.rs | LO phase offset estimation | Corrects hardware-induced phase corruption |
coherence.rs | Z-score coherence scoring | Statistical test on EM field stability |
coherence_gate.rs | Accept/Reject decisions | Quality control on EM measurements |
adversarial.rs | Physical impossibility detection | Enforces EM consistency constraints |
Differentiable ray-tracing forward model: Train a neural surrogate on ray-tracing simulations of CSI for various body poses in the deployment room. Use as physics constraint in pose estimation.
Fresnel zone integration: Augment the attention mechanism in multistatic.rs with Fresnel zone geometry -- links where the body falls within the first Fresnel zone should receive higher attention weight.
xPRA-LM inverse scattering: For higher-resolution body imaging than RTI, implement the Extended Phaseless Rytov Approximation. Our tomography module currently uses the simpler additive attenuation model.
HoloCSI-style sparse transformer: Replace the dense attention in cross-viewpoint fusion with top-k sparse attention for efficiency on ESP32-constrained deployments.
Physics-informed training loss: When training the DensePose model, add a loss term penalizing physically impossible CSI patterns (e.g., signals that would require faster-than-light propagation or negative attenuation).