Modeling stochastic differential equations (SDEs) is crucial for understanding complex dynamical systems in various scientific fields. Recent methods often employ neural network-based models, which typically represent SDEs through a combination of deterministic and stochastic terms. However, these models usually lack interpretability and have difficulty generalizing beyond their training domain. This paper introduces the Finite Expression Method (FEX), a symbolic learning approach designed to derive interpretable mathematical representations of the deterministic component of SDEs. For the stochastic component, we integrate FEX with advanced generative modeling techniques to provide a comprehensive representation of SDEs. The numerical experiments on linear, nonlinear, and multidimensional SDEs demonstrate that FEX generalizes well beyond the training domain and delivers more accurate long-term predictions compared to neural network-based methods. The symbolic expressions identified by FEX not only improve prediction accuracy but also offer valuable scientific insights into the underlying dynamics of the systems, paving the way for new scientific discoveries.

Preprint submitted to Elsevier March 10, 2025 algorithms to solve such inverse-type problems advance different fields including inferring neural circuit dynamics from spiking data [42] in neuroscience, modeling and predicting complex weather patterns from historical data [9] in climate science, uncovering disease transmission dynamics from infection case counts over time [46] in epidemiology, and deducing reaction rates from experimental concentration-time profiles in reaction kinetics in biochemistry [30]. However, such inverse-type problems pose substantial mathematical and computational challenges, particularly when data are limited and noisy, motivating ongoing research into novel algorithms and theoretical frameworks to improve models' reconstruction accuracy and efficiency. In this paper, we study the inverse problem of inferring the distribution of model parameters for several dynamical systems including ordinary differential equations (ODEs), partial differential equations (PDEs), and stochastic differential equations (SDEs) from time-series data or spatiotemporal data. Existing methods for such problems can be broadly categorized into traditional statistical approaches and modern data-driven techniques. Traditional statistical methods often involve parameter estimation frameworks. For example, linear and nonlinear regression methods play a role in simpler systems where the functional form of the model is partially known [13]. Furthermore, maximum likelihood estimation and Bayesian inference methods [16, 33] are often adopted. Maximum likelihood estimation optimizes the likelihood of model parameter values in a proposed model from observed data, while Bayesian methods incorporate prior information and compute posterior distributions. These approaches are widely used in applications such as reaction network reconstruction and epidemiological modeling.

This study introduces a training-free conditional diffusion model for learning unknown stochastic differential equations (SDEs) using data. The proposed approach addresses key challenges in computational efficiency and accuracy for modeling SDEs by utilizing a score-based diffusion model to approximate their stochastic flow map. Unlike the existing methods, this technique is based on an analytically derived closed-form exact score function, which can be efficiently estimated by Monte Carlo method using the trajectory data, and eliminates the need for neural network training to learn the score function. By generating labeled data through solving the corresponding reverse ordinary differential equation, the approach enables supervised learning of the flow map. Extensive numerical experiments across various SDE types, including linear, nonlinear, and multi-dimensional systems, demonstrate the versatility and effectiveness of the method. The learned models exhibit significant improvements in predicting both short-term and long-term behaviors of unknown stochastic systems, often surpassing baseline methods like GANs in estimating drift and diffusion coefficients.

Learning unknown stochastic differential equations (SDEs) from observed data is a significant and challenging task with applications in various fields. Current approaches often use neural networks to represent drift and diffusion functions, and construct likelihood-based loss by approximating the transition density to train these networks. However, these methods often rely on one-step stochastic numerical schemes, necessitating data with sufficiently high time resolution. In this paper, we introduce novel approximations to the transition density of the parameterized SDE: a Gaussian density approximation inspired by the random perturbation theory of dynamical systems, and its extension, the dynamical Gaussian mixture approximation (DynGMA). Benefiting from the robust density approximation, our method exhibits superior accuracy compared to baseline methods in learning the fully unknown drift and diffusion functions and computing the invariant distribution from trajectory data. And it is capable of handling trajectory data with low time resolution and variable, even uncontrollable, time step sizes, such as data generated from Gillespie's stochastic simulations. We then conduct several experiments across various scenarios to verify the advantages and robustness of the proposed method.

Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an alternative approach, utilizing neural networks combined with ODE solvers to learn continuous latent representations through parameterized vector fields. Neural Stochastic Differential Equations (Neural SDEs) extend Neural ODEs by incorporating a diffusion term, although this addition is not trivial, particularly when addressing irregular intervals and missing values. Consequently, careful design of drift and diffusion functions is crucial for maintaining stability and enhancing performance, while incautious choices can result in adverse properties such as the absence of strong solutions, stochastic destabilization, or unstable Euler discretizations, significantly affecting Neural SDEs' performance. In this study, we propose three stable classes of Neural SDEs: Langevin-type SDE, Linear Noise SDE, and Geometric SDE. Then, we rigorously demonstrate their robustness in maintaining excellent performance under distribution shift, while effectively preventing overfitting. To assess the effectiveness of our approach, we conduct extensive experiments on four benchmark datasets for interpolation, forecasting, and classification tasks, and analyze the robustness of our methods with 30 public datasets under different missing rates. Our results demonstrate the efficacy of the proposed method in handling real-world irregular time series data.

Stochastic differential equations (SDEs) are an important framework to model dynamics with randomness, as is common in most biological systems. The inverse problem of integrating these models with empirical data remains a major challenge. Here, we present a software package, PyDaDDy (Python Library for Data Driven Dynamics) that takes time series data as an input and outputs an interpretable SDE. We achieve this by combining traditional approaches from stochastic calculus literature with state-of-the-art equation discovery techniques. We validate our approach on synthetic datasets, and demonstrate the generality and applicability of the method on two real-world datasets of vastly different spatiotemporal scales: (i) collective movement of fish school where stochasticity plays a crucial role, and (ii) confined migration of a single cell, primarily following a relaxed oscillation. We make the method available as an easy-to-use, open-source Python package, PyDaddy (Python Library for Data Driven Dynamics).

Time-series data are ubiquitous in the real world, often comprising a series of data points recorded at varying time intervals. Understanding the underlying structures between variables associated with temporal processes is of paramount importance for numerous real-world applications (Spirtes et al., 2000; Berzuini et al., 2012; Peters et al., 2017). Although randomised experiments are considered the gold standard for unveiling such relationships, they are frequently hindered by factors such as cost and ethical concerns. Structure learning seeks to infer hidden structures from purely observational data, offering a powerful approach for a wide array of applications (Bellot et al., 2021; Löwe et al., 2022; Runge, 2018; Tank et al., 2021; Pamfil et al., 2020; Gong et al., 2022). However, many existing structure learning methods for time series are inherently discrete, assuming that the underlying temporal processes are discretized in time and requiring uniform sampling intervals throughout the entire time range. Consequently, these models face two key limitations: (i) they may misrepresent the true underlying process when it is continuous in time, potentially leading to incorrect inferred relationships; and (ii) they struggle with handling irregular sampling intervals, which frequently arise in fields such as biology (Trapnell et al., 2014; Qiu et al., 2017; Qian et al., 2020) and climate science (Bracco et al., 2018; Raia, 2008). Although there exists a previous work (Bellot et al., 2021) that also tries to infer the underlying structure from the continuous-time perspective, its framework based on ordinary differential equations (ODE)

Uncertainty quantification is a critical yet unsolved challenge for deep learning, especially for the time series imputation with irregularly sampled measurements. To tackle this problem, we propose a novel framework based on the principles of recurrent neural networks and neural stochastic differential equations for reconciling irregularly sampled measurements. We impute measurements at any arbitrary timescale and quantify the uncertainty in the imputations in a principled manner. Specifically, we derive analytical expressions for quantifying and propagating the epistemic and aleatoric uncertainty across time instants. Our experiments on the IEEE 37 bus test distribution system reveal that our framework can outperform state-of-the-art uncertainty quantification approaches for time-series data imputations.

Modelling joint dynamics of liquid vanilla options is crucial for arbitrage-free pricing of illiquid derivatives and managing risks of option trade books. This paper develops a nonparametric model for the European options book respecting underlying financial constraints and while being practically implementable. We derive a state space for prices which are free from static (or model-independent) arbitrage and study the inference problem where a model is learnt from discrete time series data of stock and option prices. We use neural networks as function approximators for the drift and diffusion of the modelled SDE system, and impose constraints on the neural nets such that no-arbitrage conditions are preserved. In particular, we give methods to calibrate \textit{neural SDE} models which are guaranteed to satisfy a set of linear inequalities. We validate our approach with numerical experiments using data generated from a Heston stochastic local volatility model.

In this paper we develop numerical pricing methodologies for European style Exchange Options written on a pair of correlated assets, in a market with finite liquidity. In contrast to the standard multi-asset Black-Scholes framework, trading in our market model has a direct impact on the asset's price. The price impact is incorporated into the dynamics of the first asset through a specific trading strategy, as in large trader liquidity model. Two-dimensional Milstein scheme is implemented to simulate the pair of assets prices. The option value is numerically estimated by Monte Carlo with the Margrabe option as controlled variate. Time complexity of these numerical schemes are included. Finally, we provide a deep learning framework to implement this model effectively in a production environment.