Rotational symmetry is one of the most important structural principles in machine learning on 3D data. In applications ranging from physics and materials science to 3D computer vision, predictions should not depend on an arbitrary choice of coordinate frame. Rotational equivariance captures this requirement mathematically by enforcing that a rotation of the input induces a corresponding transformation of the model output. This tutorial provides a comprehensive introduction to rotational equivariance, starting from the physical and geometric intuition behind coordinate independence and building up the necessary machinery from geometric deep learning, group theory, and representation theory. We introduce message passing on Euclidean graphs, group actions and representations, spherical harmonics, Wigner matrices, tensor products, and Clebsch-Gordan decomposition, and explain how these ingredients give rise to modern equivariant architectures. We then survey the principal strategies for incorporating rotational equivariance in deep learning, including group convolutions, internal tensorial representations, and canonicalization-based methods, and discuss their practical strengths and limitations. The tutorial aims to lower the barrier to the subject by connecting the underlying mathematics to practical model design, by unifying ideas that are often expressed in different formal languages, and by helping practitioners choose among competing approaches through a clear discussion of their trade-offs.
Claire Vernade, Onno Eberhard, Martha White +4cs.LG
Reinforcement learning and control theory are two adjacent scientific fields that focus on optimizing the controller of unknown dynamical systems using feedback. While both fields have common roots in dynamic programming, they have evolved with distinct methodologies, goals, and cultures. Despite decades of mutual influence, a significant gap persists between the two communities. This tutorial introduces adaptive control, actor-critic reinforcement algorithms, and a new way to combine these two paradigms for data-driven decision making on a classical locomotion control problem. Our aim is to provide a foundation for understanding the core differences between the two approaches and insights to help experts in each field better understand and engage with the tools and approaches of the other.
FastLAS is a scalable system for Inductive Logic Programming (ILP): you give it some background knowledge, a language bias, and a set of examples, and it searches for a set of logic program rules (a hypothesis) that explains the examples. These notes are a hands-on introduction to writing FastLAS programs. They are organised as a programmer's guide: syntax first, then a ladder of worked, numbered examples of increasing difficulty. Every self-contained example here has been run against FastLAS 2.2.0 and shows the tool's actual output. We keep theory to the minimum needed to write correct programs; throughout, set-off notes flag where FastLAS differs from its sibling system ILASP, and where the two learning algorithms (--opl and --nopl) behave differently. The document is intended as an unofficial tutorial to FastLAS 2.2.0, not as an official language specification.
Yao Cheng Li, Ana Larrañaga, Steven L. Brunton +1cs.LG
Many engineering problems involve phenomena whose governing equations are poorly characterized or only partially known. Surrogate modeling techniques such as neural networks can capture the behavior of these systems, but they typically demand large training datasets that are difficult to obtain in engineering contexts and yield models with limited physical interpretability. The Sparse Identification of Nonlinear Dynamics (SINDy) method addresses both limitations by performing sparse regression over libraries of candidate nonlinear terms, recovering interpretable governing equations from comparatively small datasets. Although SINDy has been demonstrated extensively on canonical benchmark systems, its application to practical engineering problems is less widely documented. This tutorial introduces the SINDy method and progressively builds toward its main extensions, from noise-robust weak-form and ensembling-based variants to constrained and parametrizable formulations. The paper and the accompanying tutorial (available at https://github.com/paullililili/SINDy4Engineers) is organized in three parts: the first introduces the standard SINDy algorithm and progressively extends it, inviting readers without prior knowledge to follow each step and adapt the methods to their own problems; the remaining two parts present detailed case studies on (1) the system identification of an unmanned aerial vehicle and (2) a chaotic thermosyphon heat exchanger. Through these examples, we aim to demonstrate that SINDy is simple to implement yet flexible enough to serve as a valuable identification tool for advanced engineering applications.
These notes give a proof-oriented introduction to diffusion models from the viewpoint of sampling, tracing a single arc from classical sampling dynamics to modern diffusion samplers, their error analysis, and inference-time control. Throughout, the material is layered into core definitions and identities proved in full, representative estimates proved under simplifying assumptions, and research-level theorems stated with a proof roadmap. The intended audience is beginning graduate students with a background in probability but no prior exposure to stochastic differential equations, stochastic numerics, or diffusion models.
World models are increasingly used in embodied intelligence and generative simulation, yet their scope remains ambiguous across communities. This tutorial presents a design-space view of world models as action-conditioned predictive models that estimate the future evolution of task-relevant observations or states. We categorize existing methods into observation-space and state-space world models, comparing their trade-offs in visual fidelity, spatial structure, physical interpretability, and control usability. We further introduce world action models, which connect predicted futures with executable robot actions, and summarize four representative paradigms: imagine-then-execute, video-feature-conditioned action prediction, joint video-action modeling, and auxiliary video prediction for policy learning. The goal of this tutorial is to clarify the conceptual scope of world (action) models and provide a structured taxonomy for embodied prediction and control.
This tutorial paper provides a step-by-step, reproducible walkthrough of NeuraDock Agent, an open-source EEG agent focused on Alpha dynamics and visual cognitive-load analysis. The goal is practical: a reader should be able to install the agent, run EEG preprocessing and quality control, generate Alpha dynamics figures, perform within-subject Rest/Task visual cognitive-load comparison, run the public mini-dataset analyses and compare them with the reference validation summary, start an online dashboard, call the real-time API from an external application, and use the LLM interpretation layer to explain quality risks. Existing EEG toolkits provide excellent offline analysis, but assembling a real-time, quality-gated cognitive-load pipeline often requires manually bridging acquisition, custom QC, Alpha feature extraction, and a web API; this tutorial closes that offline-to-online gap. The tutorial uses a quality-gated workflow: downstream Alpha and workload metrics are computed only after preprocessing and QC gating rather than directly from raw EEG. In the included mini-dataset validation, the agent processed 18 recordings, generated 10 within-subject comparisons, observed task-related posterior Alpha suppression in 7 of 10 contrasts, estimated initial evidence of within-subject repeatability, and benchmarked local online API latency. The tutorial is intended for researchers, developers, and applied teams who want a transparent path from EEG files to real-time visual cognitive-load prototypes.
Plugging predictions of unknown parameters into downstream optimization problems, often referred to as the ``predict-then-optimize'' paradigm, has long been a standard approach in decision-making under uncertainty. However, improved predictive accuracy does not, in general, translate into improved decision quality. This disconnect has motivated growing interest in decision-focused learning (DFL) within the operations research community. This tutorial reviews recent developments in DFL and highlights key methodological insights, with a particular focus on stochastic linear programming as the downstream decision-making problem. We discuss why several widely used tools in traditional statistical learning are not directly suited to decision-focused settings and must be rethought, including (i) data collection strategies driven purely by predictive uncertainty and (ii) distributional distance measures such as the Wasserstein distance. We summarize properties of DFL that distinguish it from conventional predictive modeling and provide insights into the development of new decision-focused tools.
Text-to-image generation is a form of generative artificial intelligence (GenAI) that converts textual descriptions into images. Most applications of GenAI in modeling and simulation (M&S) have focused on large language models for documentation, coding, or explanation. By contrast, the potential of image generation remains largely unexplored. This tutorial introduces text-to-image generation to the M&S community and details how it can support several M&S tasks, including communicating conceptual models, visualizing simulation outcomes, generating educational materials, and interfacing heterogeneous models in multi-scale simulations. The tutorial combines conceptual guidance with practical workflows, explaining how modern image generators operate, how prompts and simulation outputs can be translated into visual scenes, and how practitioners can integrate these tools into reproducible local pipelines. By focusing on transferable principles rather than specific tools, the tutorial equips M&S practitioners with the knowledge needed to evaluate, adopt, and adapt text-to-image generation in their simulation workflows.
Luca Martino, Fernando Llorentestat.CO astro-ph.CO astro-ph.IM stat.ML
The nested sampling (NS) technique has gained widespread attention, particularly in cosmology and astronomy, due to its ability to efficiently explore high-likelihood regions - a feature akin to an implicit likelihood optimization that underlies its success. While the full theoretical derivation of NS is complex and involves several approximations, the central challenge lies in sampling from the likelihood-constrained priors, which is crucial for its performance. This work provides a comprehensive and detailed exposition of NS derivation, clarifying both its theoretical foundations and practical challenges. We provide a thorough description of the NS procedure, emphasizing both its strengths and potential limitations. In doing so, this work seeks to deepen understanding of the method and to foster the development of future enhancements, novel variants, and more efficient implementations across a wide range of scientific applications. Thus, the main contribution of this work is twofold: it serves both as a tutorial for newcomers to the field and as a critical review for experienced practitioners.
This tutorial develops diffusion models from the viewpoint of differential equations. We begin with the conditional Gaussian forward process and show that this path admits both an ordinary differential equation (ODE) representation and a stochastic differential equation (SDE) representation. Averaging the conditional process over the data distribution then yields marginalized forward ODE and SDE formulations that transport the data distribution $p_0=p_{\mathrm{data}}$ to a Gaussian prior $p_1=\mathcal{N}(0,I)$. We next derive the corresponding reverse-time dynamics, namely the reverse SDE and the reverse probability-flow ODE, both of which are governed by the marginal score $\grad\log p_t(x)$. This leads to a training objective for score estimation and shows that the standard noise-prediction objective is equivalent to score matching up to an additive constant independent of the model parameters. We then discuss sampling methods for the learned reverse dynamics, including DPM-Solver, as well as guided sampling through classifier guidance and classifier-free guidance. Finally, we compare DDPM and DDIM with the reverse SDE/ODE framework and show that they share the same training objective, while DDPM sampling corresponds to discrete reverse-SDE sampling and DDIM sampling corresponds to reverse-ODE sampling.