Anomalous diffusion : from life to machines

Diffusion refers to numerous phenomena, by which particles and bodies of all kinds move throughout any kind of material, has emerged as one of the most prominent subjects in the study of complex systems. Motivated by the recent developments in experimental techniques, the field had an important burst in theoretical research, particularly in the study of the motion of particles in biological environments. Just with the information retrieved from the trajectories of particles we are now able to characterize many properties of the system with astonishing accuracy. For instance, when Einstein introduced the diffusion theory back in 1905, he used the motion of microscopic particles to calculate the size of the atoms of the liquid these were suspended. Initially, most of the experimental evidence showed that such systems follow Brownian-like dynamics, i.e. the homogeneous interaction between the particles and the environment led to its stochastic, but uncorrelated motion. However, we know now that such a simple explanation lacks crucial phenomena that have been shown to arise in a plethora of physical systems. The divergence from Brownian dynamics led to the theory of anomalous diffusion, in which the particles are affected in a way or another by their interactions with the environment such that their diffusion changes drastically. For instance features such as ergodicity, Gaussianity, or ageing are now crucial for in the understanding of diffusion processes, well beyond Brownian motion. In theoretical terms, anomalous diffusion has a well-developed framework, able to explain most of the current experimental observations. However, it has been usually focused in describing the systems in terms of its macroscopic behaviour. This means that the processes are described by means of general models, able to predict the average or collective features. Even though such an approach leads to a correct description of the system and hints on the actual underlying phenomena, it lacks the understanding of the particular microscopic interactions leading to anomalous diffusion. The work presented in this Thesis has two main goals. First, we will explore how one may use microscopical (or phenomenological) models to understand anomalous diffusion. By microscopical model we refer to a model in which we will set exactly how the interactions between the various components of a system are. Then, we will explore how these interactions may be tuned in order to recover and control anomalous diffusion and how its features depend on the properties of the system. We will explore crucial topics arising in recent experimental observations, such as weak-ergodicity breaking or liquid-liquid phase separation. Second, we will survey the topic of trajectory characterization. Even if our theories are extremely well developed, without an accurate tool for studying the trajectories observed in experiments, we will be unable to correctly make any faithful prediction. In particular, we will introduce one of the first machine learning techniques that can be used for such purpose, even in systems where previous techniques failed largely.La difusión es el fenómeno por el cual partículas de todas formas y tamaños se mueven a través del entorno que les rodea. Su estudio se ha convertido en una potente herramienta para entender el comportamiento de sistemas complejos. Gracias al reciente desarrollo de diferentes técnicas experimentales, este fenómeno ha generado un enorme interés tanto desde el punto de vista experimental como del teórico, y en especial,en el estudio del movimiento de partículas microscópicas en entornos biológicos. Mediante el análisis de las trayectorias de estas partículas, no solo somos capaces de caracterizar sus propiedades, sino también las de su entorno. El propio Albert Einstein, autor junto con Marian Smoluchowski de la teoría de la difusión, demostró que era posible calcular el radio de los átomos de un líquido simplemente mediante el análisis del movimiento de una partícula suspendida en este. Esta teoría, que dio origen a lo que hoy conocemos como movimiento Browniano, consideraba que la interacción homogénea de una partícula con su entorno provocaba el movimiento aleatorio de esta última. Aunque el movimiento Browniano haya sido utilizado para describir una enorme cantidad de experimentos, hoy sabemos que existen sistemas particulares que se desvían de sus predicciones. Esta divergencia ha dado pie al desarrollo de la teoría de la difusión anómala, en la que, debido a las propiedades de las partículas y sus entornos, la difusión difiere drásticamente de las predicciones de la teoría Browniana. Algunos fenómenos como la ergodicidad, Gausianidad o el envejecimiento de difusión, particulares de la difusión anómala, son hoy en día cruciales para entender el movimiento de partículas en sistemas complejos. En términos teóricos, la difusión anómala tiene unas bases firmes, con las cuáles se explica gran parte de las observaciones experimentales más recientes. Esta teoría, sin embargo, suele centrarse en la descripción de la difusión desde un punto de vista macroscópico. Esto quiere decir: analizar un sistema mediante modelos generales, capaces de predecir propiedades colectivas o globales. Aunque las teorías macroscópicas consiguen describir correctamente la mayoría de los procesos de difusión, no tienen la capacidad de discernir qué tipo de interacciones dan lugar a la difusión anómala. El trabajo presentado en esta tesis tiene dos objetivos principales. El primero es explorar el uso de modelos microscópicos (o fenomenológicos) para entender la difusión anómala. Un modelo microscópico, en contraposición al macroscópico, describe el sistema a partir de sus propiedades específicas. En este caso, a partir del tipo de interacciones que existen entre las partículas y su entorno. El objetivo es por lo tanto entender cuáles de estas interacciones producen difusión anómala. Además, caracterizaremos los parámetros macroscópicos de la difusión, como el exponente anómalo, y mostraremos como depende de las propiedades del sistema. En el camino, exploraremos cómo fenómenos como la rotura débil de la ergodicidad (weak-ergodicity breaking) o la separación de fase aparecen en sistemas con interacciones complejas. El segundo objetivo consiste en el desarrollo de técnicas para la caracterización de trayectorias provenientes de procesos de difusión. Aunque nuestro entendimiento teórico llegue a niveles insospechados en los próximos años, sin un análisis correcto y preciso de las trayectorias experimentales, jamás podremos construir un puente entre teoría y experimentos. Por tanto, el desarrollo de técnicas con las que analizar con la mayor precisión posible dichas trayectorias es un problema igual de importante que el desarrollo teórico de la difusión. En este trabajo, estudiaremos cómo las técnicas de aprendizaje automático (Machine Learning) pueden ser utilizadas para caracterizar dichas trayectorias, llegando a niveles de precisión y análisis muy por encim

Certificates of quantum many-body properties assisted by machine learning

Computationally intractable tasks are often encountered in physics and optimization. Such tasks often comprise a cost function to be optimized over a so-called feasible set, which is specified by a set of constraints. This may yield, in general, to difficult and non-convex optimization tasks. A number of standard methods are used to tackle such problems: variational approaches focus on parameterizing a subclass of solutions within the feasible set; in contrast, relaxation techniques have been proposed to approximate it from outside, thus complementing the variational approach by providing ultimate bounds to the global optimal solution. In this work, we propose a novel approach combining the power of relaxation techniques with deep reinforcement learning in order to find the best possible bounds within a limited computational budget. We illustrate the viability of the method in the context of finding the ground state energy of many-body quantum systems, a paradigmatic problem in quantum physics. We benchmark our approach against other classical optimization algorithms such as breadth-first search or Monte-Carlo, and we characterize the effect of transfer learning. We find the latter may be indicative of phase transitions, with a completely autonomous approach. Finally, we provide tools to generalize the approach to other common applications in the field of quantum information processing.Comment: 22 pages (12.5 + appendices), 8 figure

Optimal foraging strategies can be learned

The foraging behavior of animals is a paradigm of target search in nature. Understanding which foraging strategies are optimal and how animals learn them are central challenges in modeling animal foraging. While the question of optimality has wide-ranging implications across fields such as economy, physics, and ecology, the question of learnability is a topic of ongoing debate in evolutionary biology. Recognizing the interconnected nature of these challenges, this work addresses them simultaneously by exploring optimal foraging strategies through a reinforcement learning framework. To this end, we model foragers as learning agents. We first prove theoretically that maximizing rewards in our reinforcement learning model is equivalent to optimizing foraging efficiency. We then show with numerical experiments that, in the paradigmatic model of non-destructive search, our agents learn foraging strategies which outperform the efficiency of some of the best known strategies such as L\'evy walks. These findings highlight the potential of reinforcement learning as a versatile framework not only for optimizing search strategies but also to model the learning process, thus shedding light on the role of learning in natural optimization processes.Comment: 12 pages (6 main text) and 9 figures. Codes and tutorials available at: https://gorkamunoz.github.io/rl_opts

Learning minimal representations of stochastic processes with variational autoencoders

Stochastic processes have found numerous applications in science, as they are broadly used to model a variety of natural phenomena. Due to their intrinsic randomness and uncertainty, they are however difficult to characterize. Here, we introduce an unsupervised machine learning approach to determine the minimal set of parameters required to effectively describe the dynamics of a stochastic process. Our method builds upon an extended $\beta$-variational autoencoder architecture. By means of simulated datasets corresponding to paradigmatic diffusion models, we showcase its effectiveness in extracting the minimal relevant parameters that accurately describe these dynamics. Furthermore, the method enables the generation of new trajectories that faithfully replicate the expected stochastic behavior. Overall, our approach enables for the autonomous discovery of unknown parameters describing stochastic processes, hence enhancing our comprehension of complex phenomena across various fields.Comment: 9 pages, 5 figures, 1 table. Code available at https://github.com/GabrielFernandezFernandez/SPIVA

Inferring pointwise diffusion properties of single trajectories with deep learning

In order to characterize the mechanisms governing the diffusion of particles in biological scenarios, it is essential to accurately determine their diffusive properties. To do so, we propose a machine learning method to characterize diffusion processes with time-dependent properties at the experimental time resolution. Our approach operates at the single-trajectory level predicting the properties of interest, such as the diffusion coefficient or the anomalous diffusion exponent, at every time step of the trajectory. In this way, changes in the diffusive properties occurring along the trajectory emerge naturally in the prediction, and thus allow the characterization without any prior knowledge or assumption about the system. We first benchmark the method on synthetic trajectories simulated under several conditions. We show that our approach can successfully characterize both abrupt and continuous changes in the diffusion coefficient or the anomalous diffusion exponent. Finally, we leverage the method to analyze experiments of single-molecule diffusion of two membrane proteins in living cells: the pathogen-recognition receptor DC-SIGN and the integrin $\alpha5\beta1$. The analysis allows us to characterize physical parameters and diffusive states with unprecedented accuracy, shedding new light on the underlying mechanisms.Comment: 17 pages, 9 figures, 1 table. Code is found in https://github.com/BorjaRequena/ste

Efficient training of energy-based models via spin-glass control

[EN] We introduce a new family of energy-based probabilistic graphical models for efficient unsupervised learning. Its definition is motivated by the control of the spin-glass properties of the Ising model described by the weights of Boltzmann machines. We use it to learn the Bars and Stripes dataset of various sizes and the MNIST dataset, and show how they quickly achieve the performance offered by standard methods for unsupervised learning. Our results indicate that the standard initialization of Boltzmann machines with random weights equivalent to spin-glass models is an unnecessary bottleneck in the process of training. Furthermore, this new family allows for very easy access to low-energy configurations, which points to new, efficient training algorithms. The simplest variant of such algorithms approximates the negative phase of the log-likelihood gradient with no Markov chain Monte Carlo sampling costs at all, and with an accuracy sufficient to achieve good learning and generalization.ML and AA groups acknowledge the Spanish Ministry MINECO and State Research Agency AEI (FIDEUA PID2019-106901GBI00/10.13039/501100011033, Severo Ochoa Grant Nos. SEV-2015-0522 and CEX2019-000910-S, FPI), the European Social Fund, Fundacio Cellex, Fundacio Mir-Puig, Generalitat de Catalunya (AGAUR Grant Nos. 2017 SGR 1341 and SGR 1381, CERCA program, QuantumCAT U16-011424, co-funded by ERDF Operational Program of Catalonia 2014-2020), ERC AdG NOQIA and CERQUTE, EU FEDER, MINECO-EU QUANTERA MAQS (funded by the State Research Agency AEI PCI2019-111828-2/10.13039/501100011033), the National Science Centre, Poland-Symfonia Grant No. 2016/20/W/ST4/00314 and the AXA Chair in Quantum Information Science. A P-K acknowledges funding from Fundacio Obra Socialla Caixa' (LCF/BQ/ES15/10360001) and the European Union's Horizon 2020 research and innovation programme-Grant Agreement No. 648913. G M-G acknowledges funding from Fundacio Obra Social 'la Caixa' (LCF-ICFO grant). M A G-M acknowledges funding from the Spanish Ministry of Education and Vocational Training (MEFP) through the Beatriz Galindo program 2018 (BEAGAL18/00203).Pozas-Kerstjens, A.; Muñoz-Gil, G.; Piñol, E.; Garcia March, MA.; Acín, A.; Lewenstein, M.; Grzybowski, PR. (2021). Efficient training of energy-based models via spin-glass control. Machine Learning: Science and Technology. 2(2). https://doi.org/10.1088/2632-2153/abe8070250262

Quantitative evaluation of methods to analyze motion changes in single-particle experiments

The analysis of live-cell single-molecule imaging experiments can reveal valuable information about the heterogeneity of transport processes and interactions between cell components. These characteristics are seen as motion changes in the particle trajectories. Despite the existence of multiple approaches to carry out this type of analysis, no objective assessment of these methods has been performed so far. Here, we have designed a competition to characterize and rank the performance of these methods when analyzing the dynamic behavior of single molecules. To run this competition, we have implemented a software library to simulate realistic data corresponding to widespread diffusion and interaction models, both in the form of trajectories and videos obtained in typical experimental conditions. The competition will constitute the first assessment of these methods, provide insights into the current limits of the field, foster the development of new approaches, and guide researchers to identify optimal tools for analyzing their experiments.Comment: 19 pages, 4 figure, 2 tables. Stage 1 registered report, accepted in principle in Nature Communications (https://springernature.figshare.com/articles/journal_contribution/Quantitative_evaluation_of_methods_to_analyze_motion_changes_in_single-particle_experiments_Registered_Report_Stage_1_Protocol_/24771687

Universal representation by Boltzmann machines with Regularised Axons

It is widely known that Boltzmann machines are capable of representing arbitrary probability distributions over the values of their visible neurons, given enough hidden ones. However, sampling -- and thus training -- these models can be numerically hard. Recently we proposed a regularisation of the connections of Boltzmann machines, in order to control the energy landscape of the model, paving a way for efficient sampling and training. Here we formally prove that such regularised Boltzmann machines preserve the ability to represent arbitrary distributions. This is in conjunction with controlling the number of energy local minima, thus enabling easy \emph{guided} sampling and training. Furthermore, we explicitly show that regularised Boltzmann machines can store exponentially many arbitrarily correlated visible patterns with perfect retrieval, and we connect them to the Dense Associative Memory networks.Comment: 12 pages. Updated reference

Dysregulated FOXO1 activity drives skeletal muscle intrinsic dysfunction in amyotrophic lateral sclerosis

Amyotrophic Lateral Sclerosis (ALS) is a multisystemic neurodegenerative disorder, with accumulating evidence indicating metabolic disruptions in the skeletal muscle preceding disease symptoms, rather than them manifesting as a secondary consequence of motor neuron (MN) degeneration. Hence, energy homeostasis is deeply implicated in the complex physiopathology of ALS and skeletal muscle has emerged as a key therapeutic target. Here, we describe intrinsic abnormalities in ALS skeletal muscle, both in patient-derived muscle cells and in muscle cell lines with genetic knockdown of genes related to familial ALS, such as TARDBP (TDP-43) and FUS. We found a functional impairment of myogenesis that parallels defects of glucose oxidation in ALS muscle cells. We identified FOXO1 transcription factor as a key mediator of these metabolic and functional features in ALS muscle, via gene expression profiling and biochemical surveys in TDP-43 and FUS-silenced muscle progenitors. Strikingly, inhibition of FOXO1 mitigated the impaired myogenesis in both the genetically modified and the primary ALS myoblasts. In addition, specific in vivo conditional knockdown of TDP-43 or FUS orthologs (TBPH or caz) in Drosophila muscle precursor cells resulted in decreased innervation and profound dysfunction of motor nerve terminals and neuromuscular synapses, accompanied by motor abnormalities and reduced lifespan. Remarkably, these phenotypes were partially corrected by foxo inhibition, bolstering the potential pharmacological management of muscle intrinsic abnormalities associated with ALS. The findings demonstrate an intrinsic muscle dysfunction in ALS, which can be modulated by targeting FOXO factors, paving the way for novel therapeutic approaches that focus on the skeletal muscle as complementary target tissue

Modern applications of machine learning in quantum sciences

In these Lecture Notes, we provide a comprehensive introduction to the most recent advances in the application of machine learning methods in quantum sciences. We cover the use of deep learning and kernel methods in supervised, unsupervised, and reinforcement learning algorithms for phase classification, representation of many-body quantum states, quantum feedback control, and quantum circuits optimization. Moreover, we introduce and discuss more specialized topics such as differentiable programming, generative models, statistical approach to machine learning, and quantum machine learning.Comment: 268 pages, 87 figures. Comments and feedback are very welcome. Figures and tex files are available at https://github.com/Shmoo137/Lecture-Note