He will also present an entropy-respecting forward–backward learning scheme that mitigates the inherent ill-posedness of stochastic learning problems.

Join us for what promises to be a very insightful session!

In this talk, Arthur Bizzi will introduce Neural Kolmogorov Equations, a deterministic and parallelizable framework for learning continuous-time stochastic processes using Forward and Backward Kolmogorov Equations.

Abstract:
Learning differential equations becomes substantially more challenging in the presence of stochasticity, as Neural SDEs typically require expensive, sequential integration during training.

P11] (submitted to The Journal of Chemical Physics)
chemrxiv.org/engage/chemr...

Kooplearn library:
kooplearn.readthedocs.io/latest/

For the longer version of the thread, you can take a look at this blog post:
vladi-iit.github.io/posts/2024-1...

Publications:
[P1] NeurIPS 2022
arxiv.org/abs/2205.14027

[P2] NeurIPS2023
arxiv.org/abs/2302.02004

[P3] ICML2024
arxiv.org/abs/2312.13426

[P4] NeurIPS2023
arxiv.org/abs/2306.04520

[P5] ICLR 2024
arxiv.org/abs/2307.09912

[P6] NeurIPS2024
arxiv.org/abs/2405.12940

14/ Looking ahead, we’re excited to tackle new challenges:
• Learning from partial observations
• Modeling non-time-homogeneous dynamics
• Expanding applications in neuroscience, genetics, and climate modeling

Stay tuned for groundbreaking updates from our team! 🌍

🙏 Collaborations with the Dynamic Legged Systems group led by Claudio Semini and the Atomistic Simulations group led by Michele Parrinello enriched our research, resulting in impactful works like [P9, P10] and [P7, P11].

12/ This journey wouldn’t have been possible without the inspiring collaborations that shaped our work.

🌟 Special thanks to Karim Lounici from École Polytechnique, whose insights were a major driving force behind many projects.

11/ One of our most exciting results:
[P8] NeurIPS 2024 proposed Neural Conditional Probability (NCP) to efficiently learn conditional distributions. It simplifies uncertainty quantification and guarantees accuracy for nonlinear, high-dimensional data.

10/ [P7] NeurIPS 2024 developed methods to discover slow dynamical modes in systems like molecular simulations. This is transformative for studying rare events and costly data acquisition scenarios in atomistic systems.

9/ Addressing continuous dynamics:
[P6] NeurIPS 2024 introduced a physics-informed framework for learning Infinitesimal Generators (IG) of stochastic systems, ensuring robust spectral estimation.

8/ 🌟 Representation learning takes center stage in:
[P5] ICLR 2024
We combined neural networks with operator theory via Deep Projection Networks (DPNets). This approach enhances robustness, scalability, and interpretability for dynamical systems.

7/ 📈 Scaling up:
[P4] NeurIPS 2023 introduced a Nyström sketching-based method to reduce computational costs from cubic to almost linear without sacrificing accuracy. Validated on massive datasets like molecular dynamics, see figure.

6/ [P3] ICML 2024 addressed a critical issue in TO-based modeling: reliable long-term predictions.
Our Deflate-Learn-Inflate (DLI) paradigm ensures uniform error bounds, even for infinite time horizons. This method stabilized predictions in real-world tasks; see the figure.

5/ [P2] NeurIPS 2023 advanced TOs with theoretical guarantees for spectral decomposition—previously lacking finite sample guarantees. We developed sharp learning rates, enabling accurate, reliable models for long-term system behavior.

4/ 🔑 The journey began with:
[P1] NeurIPS 2022
We introduced the first ML formulation for learning TO, which led to the development of the open-source Kooplearn library. This step laid the groundwork for exploring the theoretical limits of operator learning from finite data.

3/TOs describe system evolution over finite time intervals, while IGs capture instantaneous rates of change. Their spectral decomposition is key for identifying dominant modes and understanding long-term behavior in complex or stochastic systems.

2/ 🌐 Our work revolves around Markov/Transfer Operators (TO) and their Infinitesimal Generators (IG)—tools that allow us to model complex dynamical systems by understanding their evolution in higher-dimensional spaces. Here’s why this matters.