Yubin Lu

Yubin Lu, Zhongjian Wang, Guillaume Bal

This paper concerns the mathematical analyses of the diffusion model in machine learning. The drift term of the backward sampling process is represented as a conditional expectation involving the data distribution and the forward diffusion. The training process aims to find such a drift function by minimizing the mean-squared residue related to the conditional expectation. Using small-time approximations of the Green's function of the forward diffusion, we show that the analytical mean drift function in DDPM and the score function in SGM asymptotically blow up in the final stages of the sampling process for singular data distributions such as those concentrated on lower-dimensional manifolds, and are therefore difficult to approximate by a network. To overcome this difficulty, we derive a new target function and associated loss, which remains bounded even for singular data distributions. We validate the theoretical findings with several numerical examples.

Yubin Lu, Xiaofan Li, Chun Liu et al.

Learning the underlying potential energy of stochastic gradient systems from partial and noisy observations is a fundamental problem arising in physics, chemistry, and data-driven modeling. Classical approaches often rely on direct regression of governing equations or velocity fields, which can be sensitive to noise and external perturbations and may fail when observations are incomplete. In this work, we propose a structure-aware, energy-based learning framework for inferring unknown potential functions in generalized diffusion processes, grounded in the energetic variational approach. Starting from the energy-dissipation law associated with the Fokker-Planck equation, we construct loss functions based on the De Giorgi dissipation functional, which consistently couple the free energy and the dissipation mechanism of the system. This formulation avoids explicit enforcement of the governing partial differential equation and preserves the underlying variational structure of the dynamics. Through numerical experiments in one, two, and three dimensions, we demonstrate that the proposed energy-based loss exhibits enhanced robustness with respect to observation time, noise level, and the diversity and amount of available training data. These results highlight the effectiveness of energy-dissipation principles as a reliable foundation for learning stochastic diffusion dynamics from data.

Fanze Kong, Chen-Chih Lai, Yubin Lu

This paper proposes a data-driven learning framework for identifying governing laws of generalized diffusions with non-gradient components. By combining energy dissipation laws with a physically consistent penalty and first-moment evolution, we design a two-stage method to recover the pseudo-potential and rotation in the pointwise orthogonal decomposition of a class of non-gradient drifts in generalized diffusions. Our two-stage method is applied to complex generalized diffusion processes including dissipation-rotation dynamics, rough pseudo-potentials and noisy data. Representative numerical experiments demonstrate the effectiveness of our approach for learning physical laws in non-gradient generalized diffusions.

Fanze Kong, Chen-Chih Lai, Yubin Lu

Conservative-dissipative dynamics are ubiquitous across a variety of complex open systems. We propose a data-driven two-phase method, the Moment-DeepRitz Method, for learning drift decompositions in generalized diffusion systems involving conservative-dissipative dynamics. The method is robust to noisy data, adaptable to rough potentials and oscillatory rotations. We demonstrate its effectiveness through several numerical experiments.

Cheng Fang, Yubin Lu, Ting Gao et al.

Recently, extracting data-driven governing laws of dynamical systems through deep learning frameworks has gained a lot of attention in various fields. Moreover, a growing amount of research work tends to transfer deterministic dynamical systems to stochastic dynamical systems, especially those driven by non-Gaussian multiplicative noise. However, lots of log-likelihood based algorithms that work well for Gaussian cases cannot be directly extended to non-Gaussian scenarios which could have high error and low convergence issues. In this work, we overcome some of these challenges and identify stochastic dynamical systems driven by $α$-stable Lévy noise from only random pairwise data. Our innovations include: (1) designing a deep learning approach to learn both drift and diffusion coefficients for Lévy induced noise with $α$ across all values, (2) learning complex multiplicative noise without restrictions on small noise intensity, (3) proposing an end-to-end complete framework for stochastic systems identification under a general input data assumption, that is, $α$-stable random variable. Finally, numerical experiments and comparisons with the non-local Kramers-Moyal formulas with moment generating function confirm the effectiveness of our method.

Luxuan Yang, Ting Gao, Yubin Lu et al.

In this article, we employ a collection of stochastic differential equations with drift and diffusion coefficients approximated by neural networks to predict the trend of chaotic time series which has big jump properties. Our contributions are, first, we propose a model called Lévy induced stochastic differential equation network, which explores compounded stochastic differential equations with $α$-stable Lévy motion to model complex time series data and solve the problem through neural network approximation. Second, we theoretically prove that the numerical solution through our algorithm converges in probability to the solution of corresponding stochastic differential equation, without curse of dimensionality. Finally, we illustrate our method by applying it to real financial time series data and find the accuracy increases through the use of non-Gaussian Lévy processes. We also present detailed comparisons in terms of data patterns, various models, different shapes of Lévy motion and the prediction lengths.

Yang Li, Yubin Lu, Shengyuan Xu et al.

With the rapid increase of valuable observational, experimental and simulated data for complex systems, much efforts have been devoted to identifying governing laws underlying the evolution of these systems. Despite the wide applications of non-Gaussian fluctuations in numerous physical phenomena, the data-driven approaches to extract stochastic dynamical systems with (non-Gaussian) Lévy noise are relatively few so far. In this work, we propose a data-driven method to extract stochastic dynamical systems with $α$-stable Lévy noise from short burst data based on the properties of $α$-stable distributions. More specifically, we first estimate the Lévy jump measure and noise intensity via computing mean and variance of the amplitude of the increment of the sample paths. Then we approximate the drift coefficient by combining nonlocal Kramers-Moyal formulas with normalizing flows. Numerical experiments on one- and two-dimensional prototypical examples illustrate the accuracy and effectiveness of our method. This approach will become an effective scientific tool in discovering stochastic governing laws of complex phenomena and understanding dynamical behaviors under non-Gaussian fluctuations.

Yubin Lu, Yang Li, Jinqiao Duan

With the rapid development of computational techniques and scientific tools, great progress of data-driven analysis has been made to extract governing laws of dynamical systems from data. Despite the wide occurrences of non-Gaussian fluctuations, the effective data-driven methods to identify stochastic differential equations with non-Gaussian Lévy noise are relatively few so far. In this work, we propose a data-driven approach to extract stochastic governing laws with both (Gaussian) Brownian motion and (non-Gaussian) Lévy motion, from short bursts of simulation data. Specifically, we use the normalizing flows technology to estimate the transition probability density function (solution of nonlocal Fokker-Planck equation) from data, and then substitute it into the recently proposed nonlocal Kramers-Moyal formulas to approximate Lévy jump measure, drift coefficient and diffusion coefficient. We demonstrate that this approach can learn the stochastic differential equation with Lévy motion. We present examples with one- and two-dimensional, decoupled and coupled systems to illustrate our method. This approach will become an effective tool for discovering stochastic governing laws and understanding complex dynamical behaviors.

Yubin Lu, Romit Maulik, Ting Gao et al.

In this work, we propose a method to learn multivariate probability distributions using sample path data from stochastic differential equations. Specifically, we consider temporally evolving probability distributions (e.g., those produced by integrating local or nonlocal Fokker-Planck equations). We analyze this evolution through machine learning assisted construction of a time-dependent mapping that takes a reference distribution (say, a Gaussian) to each and every instance of our evolving distribution. If the reference distribution is the initial condition of a Fokker-Planck equation, what we learn is the time-T map of the corresponding solution. Specifically, the learned map is a multivariate normalizing flow that deforms the support of the reference density to the support of each and every density snapshot in time. We demonstrate that this approach can approximate probability density function evolutions in time from observed sampled data for systems driven by both Brownian and Lévy noise. We present examples with two- and three-dimensional, uni- and multimodal distributions to validate the method.