Stochastic differential equations (SDEs) describe dynamical systems where deterministic flows, governed by a drift function, are superimposed with random fluctuations dictated by a diffusion function. The accurate estimation (or discovery) of these functions from data is a central problem in machine learning, with wide application across natural and social sciences alike. Yet current solutions are brittle, and typically rely on symbolic regression or Bayesian non-parametrics. In this work, we introduce FIM-SDE (Foundation Inference Model for SDEs), a transformer-based recognition model capable of performing accurate zero-shot estimation of the drift and diffusion functions of SDEs, from noisy and sparse observations on empirical processes of different dimensionalities. Leveraging concepts from amortized inference and neural operators, we train FIM-SDE in a supervised fashion, to map a large set of noisy and discretely observed SDE paths to their corresponding drift and diffusion functions. We demonstrate that one and the same (pretrained) FIM-SDE achieves robust zero-shot function estimation (i.e. without any parameter fine-tuning) across a wide range of synthetic and real-world processes, from canonical SDE systems (e.g. double-well dynamics or weakly perturbed Hopf bifurcations) to human motion recordings and oil price and wind speed fluctuations.

Markov jump processes are continuous-time stochastic processes which describe dynamical systems evolving in discrete state spaces. These processes find wide application in the natural sciences and machine learning, but their inference is known to be far from trivial. In this work we introduce a methodology for zeroshot inference of Markov jump processes (MJPs), on bounded state spaces, from noisy and sparse observations, which consists of two components. First, a broad probability distribution over families of MJPs, as well as over possible observation times and noise mechanisms, with which we simulate a synthetic dataset of hidden MJPs and their noisy observation process. Second, a neural network model that processes subsets of the simulated observations, and that is trained to output the initial condition and rate matrix of the target MJP in a supervised way. We empirically demonstrate that one and the same (pretrained) model can infer, in a zero-shot fashion, hidden MJPs evolving in state spaces of different dimensionalities. Specifically, we infer MJPs which describe (i) discrete flashing ratchet systems, which are a type of Brownian motors, and the conformational dynamics in (ii) molecular simulations, (iii) experimental ion channel data and (iv) simple protein folding models. What is more, we show that our model performs on par with state-of-the-art models which are finetuned to the target datasets. Our pretrained model is available online.

Sequences of self exciting, or inhibiting, temporal events are frequent footmarks of natural phenomena: Earthquakes are known to be temporally clustered as aftershocks are commonly triggered following the occurrence of a main event [Ogata, 1988]; in social networks, the propagation of news can be modeled in terms of information cascades over the edges of a graph [Zhao et al., 2015]; and in neuronal activity, the occurrence of one spike may increase or decrease the probability of the occurrence of the next spike over some time period [Dayan and Abbott, 2001]. Traditionally, sequences of events in continuous time are modeled by Point processes, of which Cox processes [Cox, 1955], or doubly stochastic processes, use a stochastic process for the intensity function, which depends only on time and is not effected by the occurrences of the events. The Hawkes process [Hawkes and Oakes, 1974] extends the Cox process to capture phenomena in which the past events affects future arrivals, by introducing a memory dependence via a memory kernel. When incorporating dependence of the process on its own history, due to the superposition theorem of point process, new events will depend on either an exogenous rate, which is independent of the history, or an endogenous rate from past arrivals. This results in a branching structure, where new events that originate from previous events can be seen as "children" of the past events.

Deep neural network models represent the state-of-the-art methodologies for natural language processing. Here we build on top of these methodologies to incorporate temporal information and model how to review data changes with time. Specifically, we use the dynamic representations of recurrent point process models, which encode the history of how business or service reviews are received in time, to generate instantaneous language models with improved prediction capabilities. Simultaneously, our methodologies enhance the predictive power of our point process models by incorporating summarized review content representations. We provide recurrent network and temporal convolution solutions for modeling the review content. We deploy our methodologies in the context of recommender systems, effectively characterizing the change in preference and taste of users as time evolves. Source code is available at [1].

Quantum computing for machine learning attracts increasing attention and recent technological developments suggest that especially adiabatic quantum computing may soon be of practical interest. In this paper, we therefore consider this paradigm and discuss how to adopt it to the problem of binary clustering. Numerical simulations demonstrate the feasibility of our approach and illustrate how systems of qubits adiabatically evolve towards a solution.