Welcome everyone, and thank you for being here today. My name is Jérome Eertmans, and I will present my Ph.D. work on differentiable ray tracing for radio propagation modeling.
Let me start by providing a brief context. In wireless communications, understanding how radio waves propagate through complex environments is essential. Ray tracing is a simulation technique that traces individual ray paths from a transmitter to a receiver.
RT models the key electromagnetic interactions: reflection off surfaces, diffraction around edges, and scattering from rough surfaces.
Radio waves travel through complex environments like cities.
RT simulates individual paths between TX and RX.
Each path can undergo multiple interactions.
This enables site-specific channel modeling.
So why differentiable ray tracing? The key idea is to make the entire RT simulation pipeline differentiable, meaning we can compute gradients of any output with respect to any input.
Differentiable RT motivation bullet.
Differentiable RT motivation bullet.
Differentiable RT motivation bullet.
Differentiable RT motivation bullet.
This represents a paradigm shift: from traditional CPU-based non-differentiable RT to GPU-enabled, optimization-ready differentiable RT.
Transition to GPU-enabled, differentiable ray tracing.
These motivations come with several practical challenges, which are the central theme of my thesis.
Challenge bullet.
Challenge bullet.
Challenge bullet.
Challenge bullet.
Before diving into the contributions, let me give you an overview of my Ph.D. journey. This timeline highlights the key milestones, from my student jobs in 2020 through the start of my Ph.D. in 2021, several conferences and research stays, up to EuCAP 2026 just a few weeks ago.
The timeline of my Ph.D. journey.
Milestone 1: 2020/07 - Student job (Craeye).
Milestone 2: 2020/08 - Student job (Oestges).
Milestone 3: 2021/09 - Ph.D. start.
Milestone 4: 2022/05-06 - SITB + COST Lyon + Doctoral school.
Milestone 5: 2023/03 - EuCAP Florence (MPT).
Milestone 6: 2023/12 - Confirmation.
Milestone 7: 2024/03 - EuCAP Glasgow (Smoothing).
Milestone 8: 2024/04 - COST stay Cesena.
Milestone 9: 2024/06 - COST Helsinki + DiffeRT2d.
Milestone 10: 2024/09-12 - Long stay Bologna.
Milestone 11: 2025/01 - COST Dublin (ML).
Milestone 12: 2025/04 - EuCAP Stockholm (MLM).
Milestone 13: 2025/05 - ICMLCN Barcelona (ML).
Milestone 14: 2025/09 - COST Lille.
Milestone 15: 2026/03 - Submission to npj.
Milestone 16: 2026/04 - EuCAP Dublin (FPT).
Hide final context before contribution focus.
I will focus on three main contributions, highlighted here: the smoothing technique, the ML-based path tracing, and the Fermat Path Tracing method.
Show the typical ray tracing pipeline.
Map contributions onto the pipeline blocks.
To motivate smoothing, let us inspect discontinuities in a street-canyon example by progressively removing interactions.
Start with depth 3 and diffraction enabled.
Depth = 3, diffraction OFF
Depth = 2, diffraction OFF
Depth = 1, diffraction OFF
Discontinuity motivation bullet.
Discontinuity motivation bullet.
Discontinuity motivation bullet.
Discontinuity motivation bullet.
The same discontinuity issue appears in a minimal setup through a hard Heaviside visibility model.
Zero-gradient image from EuCAP 2024.
Zero-gradient bullet.
Zero-gradient bullet.
Zero-gradient bullet.
Introduce the smoothing formulation.
Smoothing formulation bullet.
Smoothing formulation bullet.
Smoothing formulation bullet.
Compare power maps: without smoothing (left) vs with smoothing (right).
Effect of smoothing on the power map.
Introduce a concrete optimization objective and setup.
Objective-function bullet.
Objective-function bullet.
Objective equation and optimization setup image.
Now let us move to EuCAP 2024 results: power maps and optimization behavior.
Power map and optimization visuals.
Optimization video on the results slide.
Smoothing result bullet.
Smoothing result bullet.
The second main contribution addresses a key challenge: most path candidates are invalid, yet we must check exponentially many. Here we show valid paths (red) and invalid paths (gray dashed) for orders 1 to 3.
Valid vs invalid paths visualization.
Valid-vs-invalid motivation bullet.
Valid-vs-invalid motivation bullet.
Valid-vs-invalid motivation bullet.
The ray tracing pipeline takes a scene, generates path candidates, traces each one, and post-processes to extract valid paths. The key bottleneck is that most candidates are invalid.
Show the pipeline flow.
The solution is a generative model that learns to predict valid path candidates directly, bypassing brute-force enumeration.
Generative sampler solution bullet.
Generative sampler solution bullet.
Generative sampler solution bullet.
Generative sampler solution bullet.
Let us briefly look at the ML model.
The ML-based approach achieves significant speedups while maintaining accuracy comparable to conventional RT. This work was presented at ICMLCN 2025.
Results comparison placeholder.
ML result bullet.
ML result bullet.
ML result bullet.
ML result bullet.
The third and final contribution is the Fermat Path Tracing method, presented at EuCAP 2026. The key idea is a unified convex formulation that handles both reflection and diffraction using the same parametrization.
Annotated geometry / equation.
FPT setup bullet.
FPT setup bullet.
FPT setup bullet.
FPT setup bullet.
FPT setup bullet.
FPT setup bullet.
We use a BFGS quasi-Newton solver, which is well-suited for GPU execution because we fix the number of iterations to ensure uniform kernel execution.
BFGS bullet.
BFGS bullet.
BFGS bullet.
BFGS bullet.
To introduce the final component of our approach, it is first important to recall how reverse-mode AD works. Here, we illustrate it on a simple example function with two inputs and two outputs., where each operation is represented as a node in the computational graph.
The compute the gradients, AD first performs a forward pass to compute the function values. Each intermediate variable is stored for later use in the backward pass.
To actually compute the gradients, AD then performs a backward pass, starting from the output gradients and applying the chain rule to compute the gradients for each intermediate variable.
A key insight is the use of implicit differentiation. Instead of unrolling all solver iterations through the backward pass (which costs O(K) memory), we use the implicit function theorem at the converged solution.
Implicit diff bullet.
Implicit diff bullet.
Implicit diff bullet.
Implicit diff bullet.
The implicit differentiation formula only requires the converged solution, not the full iteration history.
The FPT method was benchmarked against existing approaches. Our solver approaches the speed of the image method while supporting both reflections and diffractions in a unified framework.
FPT result bullet.
FPT result bullet.
FPT result bullet.
FPT result bullet.
FPT result bullet.
FPT result bullet.
FPT result bullet.
Looking ahead, several exciting research directions remain open. The main bottleneck remains the availability of efficient open GPU solvers.
Future direction bullet.
Future direction bullet.
Future direction bullet.
Key bottleneck warning.
All of these contributions are implemented in open-source software. DiffeRT is the full 3D library, while DiffeRT2d is a lightweight 2D version I created for prototyping and teaching.
Software cards.
Open source bullet.
Open source bullet.
Open source bullet.
Open source bullet.
Open source bullet.
Beyond the scientific contributions, I am particularly proud of several achievements: ...
Proud achievement bullet.
Proud achievement bullet.
Full list of publications during the Ph.D.
Publications card, comments about each publication.
Thank you all for your attention. I am happy to take your questions.
Discuss applying smoothing to 3D intersections and trade-offs.
Show Möller-Trumbore smoothed visualization and discuss pros/cons.
Smoothing 3D discussion bullet.
Smoothing 3D discussion bullet.
Smoothing 3D discussion bullet.
Smoothing 3D discussion bullet.
Let us briefly look at the training procedure.