Waves 2025 Conference Report: Emilio McAllister Fognini

1. Introduction:

The “Conference on Mathematics of Wave Phenomena 2025” was held at Karlsruhe Institute of Technology (KIT) in Karlsruhe, Baden-W”{u}rttemberg, Germany — about an hour away from Frankfurt by train — between the 24th and 28th of February.
Karlsruhe is an old and wonderful town dating back to the early 18th century, nestled between the Black Forest and the Rhine river, and was built by the old Margrave of Baden-Durlach, Karl-Wilhelm.
In fact, the city’s layout radially extends from the Margrave’s palace with all the streets converging at the palace, and it is where Karlsruhe gets its nickname — the fan city.
KIT is the largest research centre in Germany and was founded in 1815, and has a long list of renowned researchers who have worked there including Fritz Haber and Heinrich Hertz. The conference was chaired by Roland Schnaubel — an expert in hyperbolic and parabolic PDEs and spectral theory — and focused on developments in mathematical modelling, simulation and analysis of wave-type equations and phenomena; key fields in analysis and numerical analysis.

2. My Contribution

I was invited by an ex-UCL colleague to present my unpublished work on Integrating Traditional Numerical Analysis Solvers with Learnable Operators, in a minisymposium titled, “Intersection of Biomedical Acoustics and Machine Learning”, which was chaired by my secondary supervisor Prof. Ben Cox.
Unfortunately, I fell under the weather on the day of my Talk but I was able to present my work and it was well received by the full room I was presenting to, indicating that the focus of my work fit well within the minisymposium.
Despite being the only person at the conference researching the budding field of Neural Operators, there was a good amount of overlap of concepts and techniques with my fellow researchers in the minisymposium.
The talks by Andreas Hauptmann (Fast Fourier models in circular geometry for learned reconstructions in photoacoustic tomography) and Janek Gr”{o}hl (Tackling the photoacoustic inverse problems with semi-supervised deep learning) were of particular note as they focused on leveraging Machine Learning to accelerate or supplement traditional numerical solvers — a key area of research for me — and illuminated a new perspective on how to use ML to complement the strength of traditional techniques.

3. Conference Highlights

Thankfully, I was only under the weather for a day and so I managed to attend and listen to many fascinating talks — even if many of them required more than the 4 years of mathematics I received during my MSci.
Some of the most Interesting talks were focused on Photoacoustic Tomography (PAT) and Reconstruction, which is an area in biomedical imaging where a pulse of light causes tissue to heat up and emit wideband Ultrasound.
PAT (and PAT reconstruction) problems are difficult to model in the forward direction (from light pulse to Ultrasound detection) and even more so in the inverse/reconstruction direction (from received Ultrasound to material properties of the tissue) due to both: the need to model the photoacoustic effect (generating sound from light in a medium), and the Ultrasound transmission and scattering problems.
This is further compounded in the inverse/reconstruction direction due to compounding error from the noisy Ultrasound transducers, causing the reconstructions to be both expensive and error artifact prone.

The notable talks on this topic at Waves 2025 were:

“Self Supervised Sparce-Data Image Reconstruction for Photoacoustics” by Markus Haltmeier -This Talk focused on how to simplify PAT reconstruction by using an new sensor set up to translate a 3D problem into a 2D one, and by using ML models they developed called ‘Sparce2Inverse’ and ‘Noise2Inverse’ to either clean up the noisy data before using a classical reconstruction pipeline or to use traditional denoising techniques and then apply these models to clean up artifacts during reconstruction.

“Fast Fourier models in circular geometry for learned reconstructions in photoacoustic tomography” by Andreas Hauptmann et al. — This Talk focused on the computational difficulties which data-driven end-to-end models face and discussed how they have been building more efficient forward and adjoint models with the aim of bringing the computational cost of PAT reconstruction to be small enough for common medical use.

“Tackling the photoacoustic inverse problems with semi-supervised deep learning” by Janek Gr”{o}hl et al. — This talk focused on using Generative Adversarial Networks (GANs) in order to improve the forward model for PAT problems and to aim to make the results of digital twin experiments (computer modelling of the forward PAT problem in a real world example) look more like real experimental data.

“Physics-Informed deep learning for ultrasonic imaging” by Felix Lucka — This Talk focused on concepts close to my own projects, namely incorporating traditional Algorithms formulated on wave-physics (such as delay-and-sum or f-k migration) as differentiable layers in a ML Architecture frameworks to facilitate in creatin light-weight, data-driven approaches which can be trained end-to-end for a given imaging task while only requiring a small amount of training data.

“Correlation-informed ordered dictionary learning for imaging in scattering media” by Chrysoula Tsogka et al. — This Talk also focused on using traditional numerical analysis techniques with a data-driven component, in this case this was Sparce Dictionary Learning using an encoder-decoder network, to compute super-resolution images in a scattering problem.

There were many other fascinating talks, but these were the highlights when it came to the intersection of numerical analysis and ML techniques.

4. Final Thoughts

My time at Waves 2025 was fantastic and provided me with an excellent opportunity to interact with researchers focusing on related problems to my own and using similar ideas for architecture development, as well as a better understanding of the state of the Operator Learning and Scientific Machine Learning within specialist domains like Numerical Analysis.

My biggest take away from Waves 2025 would be that we as Machine Learning (ML) researchers should engage and interact more with domain experts to solve the complex and technical problems which traditional fields, such as modelling and biomedical imaging, face, as opposed to solving these problems on our own using increasingly larger and data-hungry models.

Or in other words, we need more Machine Learning and less Artificial Intelligence as researchers already have many existing tools which work frighteningly well, we should be focusing on improving these methods as opposed to replacing them wholesale with an end-to-end AI model — I found out that an area which many researchers were interested in was uncertainty quantification for outputs of ML and AI models.

Many of the problems which non-ML researchers and specialists are trying to solve have low amounts of training data (and in medical imaging this data is noisy as well), and thus, need solutions which are robust and can be integrated into existing frameworks without requiring an external server of GPUs to function.

Due to this, I encountered a sizable use of self-supervised or semi-supervised learning within the field in order to alleviate the lack of training data and typically a focus on either older techniques — such as dictionary learning and non-variational auto-encoders.

I think that a lot of productive and fruitful work is possible if we as ML researchers could provide domain expertise regarding techniques within ML or AI which supports the work of existing researchers and collaborate on bringing cutting edge advances in the ML and AI fields into existing fields such as numerical analysis and inverse problems.

Overall I had a great time at KIT and in Karlsruhe, it was a wonderful change of pace compared to London and a beautiful and historic city, and the conference provided me a lot of discussion time with fellow researchers and food for thought about possible research directions in the future.