Hidden Order: The Geometry of Complex Patterns
Salvatore Torquato gives a public lecture in the Simons Foundation's 2026 Randomness series, connecting the geometry of complex systems with random structures and applications across mathematics, computer science, physics and chemistry.
Physics of Optimal Transport and Schrödinger Bridges
Optimal transport is a mathematical method to define a distance between probability distributions. This is particularly useful in various domains, including physics, biology, machine learning, and economics, among others. After introducing the Optimal Transport (OT) problem at finite temperature, we show how it can be formulated as a statistical physics problem. This approach allows us to derive very efficient algorithms to effectively compute the distance between two probability distributions. The a priori unrelated Schrödinger bridge (SB) problem is presented, and it is shown to be a dynamical version of the optimal transport problem. Indeed, the Schrodinger bridge looks for the most probable path in probability distribution space, which connects two given probabilities. The Schrodinger bridge problem, originally devised for freely diffusing particles, can be generalized to the case of interacting particles. It can be formulated in terms of functional integrals over bosonic fields, which allows us to derive partial differential equations that characterize the most probable paths in probability space. CARL VAN VREESWIJK MEMORIAL LECTURE 2026. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-04-15. Recording duration: 00:55:05.
Local Deep Learning without Gradients in Asymmetric Recurrent Networks
We introduce a statistical physics framework for learning in neural architectures composed of single or interconnected asymmetric attractor networks. These systems can exhibit a manifold of global fixed points capable of implementing sophisticated input-output mappings, which we characterize analytically. Learning from extensive datasets is achieved through the stabilization of fixed points via a fully distributed and local learning process, implemented at the single-neuron level. This simple mechanism yields performance comparable to that of conventional feedforward deep neural networks trained using gradient-based methods. The effectiveness of the model stems from the dense and accessible manifolds of stable fixed points, which encode the internal representations of data. Unlike other approaches to deep learning without backpropagation, our method does not attempt to estimate gradients. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2025-06-18. Recording duration: 00:45:35.
Using ML tools in neuroscience to define optimality in complex natural behavior
Biological systems must selectively encode partial information about the environment, as dictated by the capacity constraints at work in all living organisms. For example, we cannot see every feature of the light field that reaches our eyes; temporal resolution is limited by transmission noise and delays, and spatial resolution is limited by the finite number of photoreceptors and output cells in the retina. Classical efficient coding theory describes how sensory systems can maximize information transmission given such capacity constraints, but it treats all input features equally. Not all inputs are, however, of equal value to the organism. Our work quantifies whether and how the brain selectively encodes stimulus features, specifically predictive features, that are most useful for fast and effective movements. We have shown that efficient predictive computation starts at the earliest stages of the visual system in the retina. We borrow techniques from machine learning, statistical physics, and information theory to assess how we get terrific, predictive vision from these imperfect (lagged and noisy) component parts. In broader terms, we aim to build a more complete theory of efficient encoding in the brain, and along the way have found some intriguing connections between approaches to coarse graining in biology, machine learning, and physics. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-06-05. Recording duration: 00:41:40.
Matrix Factorization with Neural Networks
The factorization of a large matrix into the product of two matrices is an important mathematical problem encountered in many tasks, ranging from dictionary learning to machine learning. Statistical physics can provide on the one hand theoretical limits on the possibility of factorizing matrices in the limit of infinite size, and also practical algorithms. While this program has been successful in the case of finite rank matrices, the regime of extensive rank (scaling linearly with the dimension of the matrix) turns out to be much harder. This talk will describe a new approach to matrix factorization that maps it to neural network models of associative memory: each pattern found in the associative memory corresponds to one factor of the matrix decomposition. A detailed theoretical analysis of this new approach shows that matrix factorization in the extensive rank regime is possible when the rank is below a certain threshold. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2024-01-03. Recording duration: 00:44:31.
Understanding Machine Learning via Exactly Solvable Statistical Physics Models
The affinity between statistical physics and machine learning has a long history. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the optimization algorithms commonly used for learning.
Theory, reimagined
Physics offers countless examples for which theoretical predictions are astonishingly powerful. But it’s hard to imagine a similar precision in complex systems where the number and interdependencies between components simply prohibits a first-principles approach, look no further than the challenge of the billions of neurons and trillions of connections within our own brains. In such settings how do we even identify the important theoretical questions? We describe a systems-scale perspective in which we integrate information theory, dynamical systems and statistical physics to extract understanding directly from measurements. We demonstrate our approach with a reconstructed state space of the behavior of the nematode C. elegans, revealing a chaotic attractor with symmetric Lyapunov spectrum and a novel perspective of motor control. We then outline a maximally predictive coarse-graining in which nonlinear dynamics are subsumed into a linear, ensemble evolution to obtain a simple yet accurate model on multiple scales. With this coarse-graining we identify long timescales and collective states in the Langevin dynamics of a double-well potential, the Lorenz system and in worm behavior. We suggest that such an ``inverse’’ approach offers an emergent, quantitative framework in which to seek rather than impose effective organizing principles of complex systems.
Soft Capricious Matter: The collective behavior of particles with “noisy” interactions
Diversity in the natural world emerges from the collective behavior of large numbers of interacting objects. Statistical physics provides the framework relating microscopic to macroscopic properties. A fundamental assumption underlying this approach is that we have complete knowledge of the interactions between the microscopic entities. But what if that, even though possible in principle becomes impossible in practice ? Can we still construct a framework for describing their collective behavior ? Dense suspensions and granular materials are two often quoted examples where we face this challenge. These are systems where because of the complicated surface properties of particles there is extreme sensitivity of the interactions to particle positions. In this talk, I will present a perspective based on notions of constraint satisfaction that provides a way forward. I will focus on our recent work on the emergence of elasticity in the absence of any broken symmetry, and sketch out other problems that can be addressed using this perspective.
Biology is “messy”. So how can we take theory in biology seriously and plot predictions and experiments on the same axes?
Many of us came to biology from physics. There we have been trained on such classic examples as muon g-2, where experimental data and theoretical predictions agree to many significant digits. Now, working in biology, we routinely hear that it is messy, most details matter, and that the best hope for theory in biology is to be semi-qualitative, predict general trends, and to forgo the hope of ever making quantitative predictions with the precision that we are used to in physics. Colloquially, we should be satisfied even if data and models differ so much that plotting them on the same plot makes little sense. However, some of us won’t be satisfied by this. So can we take theory in biology seriously and predict experimental outcomes within (small) error bars? Certainly, we won’t be able to predict everything, but this is never required, even in traditional physics. But we should be able to choose some features of data that are nontrivial and interesting, and focus on them. We also should be able to find different classes of models --- maybe even null models --- that match biology better, and thus allow for a better agreement. It is even possible that large-dimensional datasets of modern high-throughput experiments, and the ensuing “more is different” statistical physics style models will make quantitative, precise theory easier. To explore the role of quantitative theory in biology, in this workshop, eight speakers will address some of the following general questions based on their specific work in different corners of biology: Which features of biological data are predictable? Which types of models are best suited to making quantitative predictions in different fields? Should theorists interested in quantitative predictions focus on different questions, not typically asked by biologists? Do large, multidimensional datasets make theories (and which theories?) more or less likely to succeed? This will be an unapologetically theoretical physics workshop — we won’t focus on a specific subfield of biology, but will explore these questions across the fields, hoping that the underlying theoretical frameworks will help us find the missing connections.
Spontaneous and driven active matter flows
Understanding individual and macroscopic transport properties of motile micro-organisms in complex environments is a timely question, relevant to many ecological, medical and technological situations. At the fundamental level, this question is also receiving a lot of attention as fluids loaded with swimming micro-organisms has become a rich domain of applications and a conceptual playground for the statistical physics of “active matter”. The existence of microscopic sources of energy borne by the motile character of these micro-swimmers is driving self-organization processes at the origin of original emergent phases and unconventional macroscopic properties leading to revisit many standard concepts in the physics of suspensions. In this presentation, I will report on a recent exploration on the question of spontaneous formation of large scale collective motion in relation with the rheological response of active suspensions. I will also present new experiments showing how the motility of bacteria can be controlled such as to extract work macroscopically.
Finding Needles in Genomic Haystacks
The ability to read the DNA sequences of different organisms has transformed biology in much the same way that the telescope transformed astronomy. And yet, much of the sequence found in these genomes is as enigmatic as the Rosetta Stone was to early Egyptologists. With the aim of making steps to crack the genomic Rosetta Stone, I will describe unexpected ways of using the physics of information transfer first developed at Bell Labs for thinking about telephone communications to try to decipher the meaning of the regulatory features of genomes. Specifically, I will show how we have been able to explore genes for which we know nothing about how they are regulated by using a combination of mutagenesis, deep sequencing and the physics of information, with the result that we now have falsifiable hypotheses about how those genes work. With those results in hand, I will show how simple tools from statistical physics can be used to predict the level of expression of different genes, followed by a description of precision measurements used to test those predictions. Bringing the two threads of the talk together, I will think about next steps in reading and writing genomes at will.
Understanding machine learning via exactly solvable statistical physics models
The affinity between statistical physics and machine learning has long history, this is reflected even in the machine learning terminology that is in part adopted from physics. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the learning algorithm.
Can machine learning learn new physics, or do we need to put it in by hand?"\
There has been a surge of publications on using machine learning (ML) on experimental data from physical systems: social, biological, statistical, and quantum. However, can these methods discover fundamentally new physics? It can be that their biggest impact is in better data preprocessing, while inferring new physics is unrealistic without specifically adapting the learning machine to find what we are looking for — that is, without the “intuition” — and hence without having a good a priori guess about what we will find. Is machine learning a useful tool for physics discovery? Which minimal knowledge should we endow the machines with to make them useful in such tasks? How do we do this? Eight speakers below will anchor the workshop, exploring these questions in contexts of diverse systems (from quantum to biological), and from general theoretical advances to specific applications. Each speaker will deliver a 10 min talk with another 10 minutes set aside for moderated questions/discussion. We expect the talks to be broad, bold, and provocative, discussing where the field is heading, and what is needed to get us there.