1. Introduction: What Is Physics-Informed AI?

Imagine you are trying to locate your phone inside a large shopping mall using radio signals. In an open field, this would be straightforward: the signal travels in a straight line, and by measuring how long it takes to arrive, you can compute the distance. But inside a building, the signal bounces off walls, ceilings, pillars, and glass surfaces. Your receiver gets hit by dozens of copies of the same signal, each arriving from a different direction and at a different time. This phenomenon is called multipath propagation.

Multipath is the single biggest enemy of accurate RF (Radio Frequency) tracking. It makes your measurements unreliable because reflected signals pretend to come from locations where the target is not. Traditional machine learning can learn patterns from data, but it does not understand the physics behind these reflections. It might happily predict that your phone teleported from the food court to the parking lot in 0.1 seconds — something that is physically impossible.

Physics-Informed AI solves this by combining the pattern-recognition power of machine learning with the hard, universal constraints of physics. The core philosophy can be stated in one sentence:

Instead of asking “What pattern fits the data?”, Physics-Informed AI asks “What explanation is physically possible AND statistically likely?”

This tutorial will walk you through every piece of this idea, from the intuition to the full mathematical derivations, with Python code and visualizations you can run yourself.

1.1 Mathematical Statement of the Problem

Before diving in, let us formally define what we are trying to solve. At each time step k, we want to estimate a state vector that describes the target:

x_k = [position_x, position_y, velocity_x, velocity_y]^T (1)

Given a sequence of noisy RF measurements:

z_k = [TOA_1, TOA_2, …, TOA_M, RSS_1, …, RSS_M]^T (2)

where TOA is Time-of-Arrival and RSS is Received Signal Strength from M anchor nodes. The goal is to find the posterior distribution:

p(x_k | z_1:k) = the probability of state x_k given ALL measurements so far (3)

This is the central mathematical object of our entire framework. Everything we build — the physics, the ML, the filters — exists to compute this distribution as accurately as possible.

2. The Multipath Problem: Why RF Tracking Is Hard

2.1 Signal Propagation in Real Environments

When a transmitter sends a radio signal, it does not travel in one straight line. In real environments, several phenomena occur simultaneously: reflection (signal bounces off walls and metal surfaces), diffraction (signal bends around edges and corners), scattering (signal splits into many directions from rough surfaces), and absorption (materials like concrete absorb signal energy). The receiver picks up a mixture of all these signal copies, and most of them are lying about where the target actually is.

Figure 1: RF signals in an urban environment. The direct path (red dashed) is blocked. The receiver only gets reflected, multi-bounce, and diffracted copies.

2.2 The Mathematics of Multipath

The received signal at the receiver can be modeled as a sum of L multipath components:

r(t) = Σ_{l=0}^{L-1} a_l · s(t – τ_l) · exp(jφ_l) + n(t) (4)

where each component l has:

• a_l = amplitude (how strong this path is)
• τ_l = time delay (how long it took to arrive)
• φ_l = phase shift (depends on path length)
• n(t) = additive noise

The direct path (l=0) gives the true distance d = c × τ_0, where c is the speed of light. But all reflected paths have τ_l > τ_0, meaning they overestimate the distance. Worse, in Non-Line-of-Sight (NLOS) conditions, the direct path may be completely blocked, leaving only reflected paths with positive bias.

2.3 The Virtual Source Method

A powerful geometric technique for analyzing reflections is the virtual source method. When a signal reflects off a flat surface, the reflected path is geometrically equivalent to a straight-line path from a “virtual source” — the mirror image of the transmitter across the reflecting surface.

Figure 2: Virtual source geometry. The reflected path TX → Wall → RX has the same length as the straight path from the virtual source TX’ to RX.

Mathematically, if the transmitter is at position (x_t, y_t) and the reflecting wall is at x = d_w, then the virtual source is at:

TX’ = (2d_w – x_t, y_t) (5)

The reflected path length equals the virtual source distance:

d_reflected = d(TX, wall) + d(wall, RX) = d(TX’, RX) (6)

This gives us the fundamental multipath delay inequality:

Δτ_l = (d_reflected,l – d_direct) / c ≥ 0 for all l (7)

This inequality is a physics constraint we will use later: no reflected path can arrive before the direct path.

2.4 Path Loss Models

The received power decreases with distance according to well-known models. The most fundamental is the Free-Space Path Loss (FSPL):

FSPL(dB) = 20 log_10(4πd / λ) (8)

where d is distance and λ = c/f is the wavelength. In real environments, the path loss exponent varies:

PL(dB) = PL_0 + 10n log_10(d/d_0) + X_σ (9)

where n is the path loss exponent (2 for free space, 3-5 for indoor), PL_0 is reference loss at distance d_0, and X_σ is a log-normal shadow fading term with standard deviation σ.

Figure 3: Left: Free-space path loss (clean theory curve) vs. real multipath measurements (scattered red points). Right: TOA-to-distance relationship showing the impossible zone (faster than light).

3. Probabilistic Inference: The Mathematical Framework

3.1 Why Probability, Not Point Estimates?

The first key insight is to stop pretending we know the exact answer. Instead of outputting a single point estimate, we maintain a full probability distribution over all possible states. This is essential because multipath creates ambiguity — the same set of measurements could have been generated by several different positions.

Figure 4: Left: Pure ML gives a wrong point estimate (fooled by multipath). Middle: Probabilistic model shows multiple candidate locations. Right: Physics constraints eliminate impossible candidates.

3.2 Bayes’ Theorem: The Foundation

Everything in this tutorial ultimately rests on Bayes’ theorem, which tells us how to update our beliefs when new evidence arrives:

p(x | z) = p(z | x) · p(x) / p(z) (10)

In our tracking context, this becomes:

p(x_k | z_1:k) ∝ p(z_k | x_k) · p(x_k | z_1:k-1) (11)

Let us carefully define each term:

Posterior p(x_k | z_1:k): What we want. The probability of the target being at state x_k given all measurements from time 1 to k. This is our best estimate of where the target is.

Likelihood p(z_k | x_k): The measurement model. If the target were at state x_k, how likely would we observe the measurement z_k? This is where ML excels — learning complex measurement models from data.

Prior p(x_k | z_1:k-1): Our prediction before seeing the new measurement. This comes from the previous posterior propagated through the motion model, and this is where physics enters — constraining what states are physically reachable.

Evidence p(z): A normalization constant. We often ignore it because it does not depend on x.

3.3 The Predict-Update Cycle

Bayesian tracking operates in a recursive two-step cycle:

Prediction Step (Physics + Motion Model)

We propagate our previous belief forward in time using the motion model:

p(x_k | z_1:k-1) = ∫ p(x_k | x_{k-1}) · p(x_{k-1} | z_1:k-1) dx_{k-1} (12)

The transition density p(x_k | x_{k-1}) encodes our physics knowledge. For example, for a constant-velocity model:

x_k = F · x_{k-1} + w_k, where w_k ~ N(0, Q) (13)

with the state transition matrix F and process noise covariance Q:

F = [[1, 0, dt, 0], [0, 1, 0, dt], [0, 0, 1, 0], [0, 0, 0, 1]]

This matrix encodes a basic physics truth: position changes by velocity times time (x_new = x_old + v * dt).

Update Step (Measurements + Likelihood)

When a new measurement z_k arrives, we update:

p(x_k | z_1:k) = p(z_k | x_k) · p(x_k | z_1:k-1) / p(z_k) (14)

The likelihood function p(z_k | x_k) for TOA measurements from M anchors is typically:

p(z_k | x_k) = ∏_{m=1}^{M} p(TOA_m | x_k) (15)

where for each anchor m at position a_m:

p(TOA_m | x_k) = N(TOA_m; ||x_k – a_m|| / c, σ_m^2) (16)

This says: the measured TOA should be close to the true distance divided by the speed of light, with some Gaussian noise. This is the LOS (Line-of-Sight) measurement model. For NLOS measurements, we need a more complex model that accounts for the positive bias from reflections.

4. Physical Laws: Mathematical Constraints for AI

4.1 Overview of Constraint Categories

Physics provides three categories of constraints that dramatically reduce the space of possible solutions. Each category has precise mathematical formulations that can be directly incorporated into our inference framework.

Figure 5: Three categories of physics constraints: speed of light (delay bounds), motion dynamics (trajectory smoothness), and geometric reflection laws.

4.2 Electromagnetic Constraints

Speed of Light Bound

The most fundamental: no signal can travel faster than light. For anchor m at position a_m:

TOA_m ≥ ||x_k – a_m|| / c (always true for valid measurements) (17)

Any measurement violating this constraint is guaranteed to be noise or interference, not a real signal. We define an indicator function:

I_SOL(TOA_m, x_k) = { 1 if TOA_m ≥ ||x_k – a_m|| / c, 0 otherwise } (18)

Multipath Delay Constraint

For any reflected path with delay τ_l:

τ_l ≥ τ_0 ≥ d_direct / c (19)

This means reflected signals always arrive AFTER the direct path. If we can identify the earliest arrival as the direct path, all later arrivals must have traveled a longer distance.

Path Loss Consistency

The received power must be consistent with the estimated distance:

|RSS_measured – RSS_predicted(d)| ≤ δ_RSS (20)

where δ_RSS is a tolerance that accounts for fading and shadowing.

4.3 Motion Dynamics Constraints

Physical objects obey Newton’s laws. We encode this as bounded velocity and acceleration:

||v_k|| = ||(x_k – x_{k-1}) / dt|| ≤ v_max (21)

||a_k|| = ||(v_k – v_{k-1}) / dt|| ≤ a_max (22)

For a pedestrian, v_max ≈ 2 m/s and a_max ≈ 3 m/s². For a vehicle, v_max ≈ 40 m/s and a_max ≈ 10 m/s². These bounds create a “reachability region” around the previous position — only states within this region have non-zero prior probability.

4.4 Geometric Constraints

Snell’s Law

Reflections must obey the law of reflection: angle of incidence equals angle of reflection:

θ_incidence = θ_reflection (23)

For a wall defined by normal vector n and a proposed reflection at point r, this means:

(TX – r) · n / ||TX – r|| = (RX – r) · n / ||RX – r|| (24)

Wall Intersection Constraint

Any proposed reflection path must physically intersect a known reflecting surface. If we have a floor plan or building map, we can verify:

I_wall(path) = { 1 if path intersects a valid surface, 0 otherwise } (25)

5. The Fusion: Combining ML and Physics

5.1 The Physics-Informed Posterior

Now we put everything together. The physics-informed posterior combines the ML likelihood with all physics constraints:

p(x_k | z_1:k) ∝ p(z_k | x_k) · p(x_k | x_{k-1}) · ∏_c I_c(x_k) (26)

where the product runs over all constraint indicator functions I_c. Expanding this:

p(x_k | z_1:k) ∝ [ML Likelihood] × [Motion Prior] × [SOL] × [Dynamics] × [Geometry] (27)

In practice, we often use soft constraints (penalty functions) instead of hard indicators to handle measurement noise:

I_c^{soft}(x) = exp(-λ_c · max(0, violation_c(x))^2) (28)

where λ_c controls how strictly the constraint is enforced. This turns each hard constraint into a Gaussian penalty that gradually pushes solutions toward physical consistency.

5.2 Bayes’ Rule Visualized Step by Step

The following figure shows Bayes’ rule operating on a 2D tracking problem with multipath. Watch how multiplying the prior (physics) by the likelihood (data) eliminates ghost peaks:

Figure 6: Bayes’ rule step by step. Step 1: Physics prior (green region = valid positions from motion + wall constraints). Step 2: ML likelihood (multiple peaks from multipath). Step 3: Multiply them — physics kills ghost peaks. Step 4: Normalize to get the final posterior.

Figure 7: Complete system architecture showing how RF measurements, physical laws, and environment maps combine through ML likelihood and physics priors into a Bayesian fusion that outputs calibrated position estimates.

6. The Kalman Filter: Optimal Linear Estimation

6.1 When Things Are Linear and Gaussian

If our motion model is linear (Eq. 13) and our measurement noise is Gaussian, Bayes’ rule has a beautiful closed-form solution: the Kalman Filter. It represents the posterior as a Gaussian distribution N(μ_k, P_k) and updates it analytically.

Prediction Equations

μ_k|k-1 = F · μ_{k-1} + B · u_k (predicted state mean) (29)

P_k|k-1 = F · P_{k-1} · F^T + Q (predicted covariance) (30)

Notice that the covariance P grows in the prediction step — uncertainty increases when we do not have new information. The process noise Q encodes how unpredictable the motion is.

Update Equations

When measurement z_k arrives with measurement matrix H and noise R:

K_k = P_k|k-1 · H^T · (H · P_k|k-1 · H^T + R)^{-1} (Kalman Gain) (31)

μ_k = μ_k|k-1 + K_k · (z_k – H · μ_k|k-1) (updated mean) (32)

P_k = (I – K_k · H) · P_k|k-1 (updated covariance) (33)

The Kalman Gain K_k is the key quantity. It automatically balances trust between the prediction and the measurement. When measurement noise R is small (high-quality sensor), K_k is large and the filter trusts the measurement. When prediction uncertainty P is small (confident model), K_k is small and the filter trusts its prediction.

Figure 8: Kalman filter mathematics visualized. Left: Prediction step grows the uncertainty ellipse. Middle: Update step fuses prediction with measurement, shrinking uncertainty. Right: Sequential estimation over time — uncertainty decreases as more measurements accumulate.

6.2 The Extended Kalman Filter (EKF)

RF tracking is inherently nonlinear because the measurement function (distance = sqrt of squared differences) is not linear. The EKF handles this by linearizing around the current estimate:

h(x) = ||x – a_m|| / c (nonlinear measurement function) (34)

H_k = ∂h/∂x |_{x=μ_k|k-1} (Jacobian matrix at predicted state) (35)

For the TOA measurement from anchor m at position a_m:

H_m = [(x – a_mx)/d_m, (y – a_my)/d_m, 0, 0] / c (36)

where d_m = ||x – a_m|| is the estimated distance. The EKF then uses this linearized H in the standard Kalman equations. This works well when the nonlinearity is mild, but can fail in severe multipath where the posterior is highly non-Gaussian (multimodal).

7. The Particle Filter: Physics-Informed AI in Action

7.1 Why Particle Filters for Multipath?

When multipath is severe, the posterior distribution becomes multimodal — there are multiple candidate positions, each corresponding to a different signal path hypothesis. The Kalman filter cannot handle this because it assumes a single Gaussian. The particle filter solves this by representing the posterior using a set of N weighted samples (particles):

p(x_k | z_1:k) ≈ Σ_{i=1}^{N} w_k^i · δ(x_k – x_k^i) (37)

where x_k^i is the position of particle i and w_k^i is its weight. This representation can approximate any distribution, no matter how complex or multimodal.

7.2 The Algorithm Step by Step

Figure 9: The four steps of a physics-informed particle filter. Random particles are scattered, weighted by measurement likelihood, checked against physics, then resampled toward the true position.

Step 1: Initialization

Distribute N particles uniformly across the valid region:

x_0^i ~ Uniform(bounds), w_0^i = 1/N for i = 1,…,N (38)

Step 2: Prediction (Propagate with Physics)

Move each particle according to the motion model with bounded noise:

x_k^i = F · x_{k-1}^i + w_k^i, subject to ||v^i|| ≤ v_max (39)

This is where motion physics enters: any particle whose proposed velocity exceeds v_max gets clipped. Particles that would end up inside a wall get rejected or reflected.

Step 3: Update (Weight by Likelihood + Physics)

Compute the weight of each particle as the product of measurement likelihood and physics constraint satisfaction:

w_k^i = p(z_k | x_k^i) · ∏_c I_c(x_k^i) (40)

Then normalize: w_k^i = w_k^i / Σ_j w_k^j. Particles that match the measurements AND satisfy physics get high weights. Particles at multipath ghost locations may match the measurements but will be penalized by physics constraints (e.g., they require an impossible velocity to reach, or their implied reflection geometry violates Snell’s law).

Step 4: Resample

Draw N new particles from the current weighted set, with probability proportional to weights:

x_k^i ~ Categorical({x_k^j, w_k^j}_{j=1}^N) (41)

After resampling, all weights are reset to 1/N. High-weight particles get duplicated; low-weight particles die off. The population naturally concentrates around the most probable position.

7.3 Python Implementation

Below is a complete, runnable Python implementation that demonstrates every concept:

Notice how each method corresponds to one step of the mathematical framework: predict (Eq. 12-13), update (Eq. 14-16 with constraints Eq. 17-18), and resample (Eq. 41). The estimate method computes the posterior mean and standard deviation from the weighted particles.

8. Physics-Informed Neural Networks (PINNs)

8.1 The PINN Loss Function

An alternative to particle filters is to train a neural network with a specially designed loss function that penalizes physics violations. This is the Physics-Informed Neural Network (PINN) approach. The total loss has three components:

L_total = L_data + α · L_physics + β · L_boundary (42)

Figure 10: The three components of the PINN loss function: data fitting (standard ML), physics satisfaction (enforce physical equations), and boundary conditions (respect physical limits).

Data Loss (Standard ML)

Measures how well the network predictions match observed measurements:

L_data = (1/N) Σ_{i=1}^{N} || ŷ_i – y_i ||^2 (43)

where ŷ_i is the network’s predicted position and y_i is the measured position (or derived from TOA/RSS measurements).

Physics Loss (The Key Innovation)

Penalizes violations of known physical equations. For RF propagation:

L_physics = (1/M) Σ_{j=1}^{M} || F(x̂_j) ||^2 (44)

where F encodes a physics residual. Examples:

• Speed of light: F = d̂ – c · TOA (distance must equal speed times time)
• Wave equation: F = ∇²E – με ∂²E/∂t² (Maxwell’s equations)
• Newton’s law: F = m·â – F_applied (dynamics constraint)

When L_physics = 0, the network’s predictions perfectly satisfy the physical laws. The weight α controls the trade-off: larger α means stricter physics enforcement.

Boundary Loss

Enforces hard physical boundaries:

L_boundary = (1/K) Σ_{k=1}^{K} max(0, violation_k)^2 (45)

Examples include: speed cannot exceed v_max, position must be within valid region, reflection angle must equal incidence angle.

8.2 Training Procedure

The PINN is trained by minimizing L_total using standard gradient descent. The physics and boundary loss terms act as regularizers that prevent the network from learning physically impossible patterns. Critically, the physics loss can be evaluated at ANY point in space-time, not just at measurement locations. This means physics provides supervision even where we have no data — dramatically reducing the need for labeled training examples.

The gradient of the total loss decomposes as:

∇L_total = ∇L_data + α∇L_physics + β∇L_boundary (46)

This means the gradient always pushes toward both data consistency AND physical consistency simultaneously. The hyperparameters α and β are typically tuned via cross-validation or set according to the relative scale of each loss.

9. Results: Pure ML vs Physics-Informed AI

9.1 Visual Tracking Comparison

The following figure shows a direct comparison of tracking the same circular trajectory. The Pure ML tracker (left) produces teleportation glitches where it locks onto multipath ghost positions. The Physics-Informed tracker (right) remains smooth because physics constraints reject impossible jumps.

Figure 11: Side-by-side tracking. Left: Pure ML with teleportation glitches from multipath. Right: Physics-Informed AI produces smooth, accurate trajectories with uncertainty ellipses.

9.2 Quantitative Scorecard

Figure 12: Performance scorecard across five dimensions.

Metric	Pure ML	Physics-Informed AI	Mathematical Reason
Accuracy	3/10	8/10	Constraints eliminate multipath ghost modes (Eq. 26)
Generalization	2/10	8/10	Physics equations are universal (F=ma holds everywhere)
Data Efficiency	2/10	7/10	L_physics evaluates at any point, not just data (Eq. 44)
Physical Consistency	1/10	9/10	Hard constraints prevent |v| > v_max (Eq. 21)
Uncertainty	2/10	9/10	Full posterior from Bayes (Eq. 11), not just point estimates

Table 1: Detailed comparison with mathematical justifications.

9.3 Error Analysis

The Root Mean Square Error (RMSE) for position estimation can be decomposed as:

RMSE = sqrt( E[||x̂ – x_true||^2] ) = sqrt( bias^2 + variance ) (47)

Pure ML suffers from high bias in NLOS conditions (multipath creates systematic errors) and high variance (no physics constraints to regularize). Physics-Informed AI reduces both: physics constraints remove the bias from ghost positions, while the Bayesian framework properly accounts for uncertainty, reducing variance.

10. Key Takeaways and Summary of Equations

This tutorial has built a complete mathematical framework for Physics-Informed AI in RF tracking. Here is a summary of the key equations and what they represent:

Equation	Name	Role in System
p(x|z) ∝ p(z|x) p(x)	Bayes’ Rule	Foundation: combine data + prior knowledge
r(t) = Σ a_l s(t-τ_l) e^{jφ_l}	Multipath Model	Describes how signals propagate
TOA ≥ d/c	Speed of Light	Hard constraint: no FTL signals
||v|| ≤ v_max	Motion Bound	Rejects impossible velocities
θ_in = θ_out	Snell’s Law	Valid reflection geometry
K = PH^T(HPH^T+R)^{-1}	Kalman Gain	Optimal linear fusion
w^i = p(z|x^i)∏I_c(x^i)	Particle Weight	Physics-informed scoring
L = L_data + α L_phys + β L_bd	PINN Loss	Neural network training

Table 2: Summary of key equations in the Physics-Informed AI framework.

“RF tracking in challenging multipath environments is enabled by fusing probabilistic inference with physical laws, allowing learning-based models to remain both data-adaptive and physically consistent.”

11. Where to Go From Here

Physics-Informed Neural Networks: Raissi, Perdikaris & Karniadakis (2019). Embed differential equations directly into neural network loss functions.

Graph Neural Networks for propagation: Model the environment as a graph where nodes are reflection points and edges are signal paths.

Variational Inference: Approximate the posterior using parametric distributions (e.g., normalizing flows) for faster inference than particle filters.

Ray tracing + ML: Use approximate ray tracing to generate candidate paths, then score them with a learned model.

Channel charting: Learn a low-dimensional representation of the radio environment that preserves spatial relationships.

Cramér-Rao Lower Bound: Derive fundamental limits on positioning accuracy in multipath channels to benchmark your system against the theoretical optimum.

The field of Physics-Informed AI for wireless communications is growing rapidly. By combining the best of machine learning with the fundamental laws of physics, we can build tracking systems that are accurate, robust, and trustworthy — even in the most challenging multipath environments.

---

I hope this tutorial will create a good foundation for you. If you want tutorials on another topic or you have any queries, please send an mail at contact@spatial-dev.guru.