Understanding the dynamic nature of biological systems is fundamental to deciphering cellular behavior, developmental processes, and disease progression. Single-cell RNA sequencing (scRNA-seq) has provided static snapshots of gene expression, offering valuable insights into cellular states at a single time point. Recent advancements in temporally resolved scRNA-seq, spatial transcriptomics (ST), and time-series spatial transcriptomics (temporal-ST) have further revolutionized our ability to study the spatiotemporal dynamics of individual cells. These technologies, when combined with computational frameworks such as Markov chains, stochastic differential equations (SDEs), and generative models like optimal transport and Schr\"odinger bridges, enable the reconstruction of dynamic cellular trajectories and cell fate decisions. This review discusses how these dynamical system approaches offer new opportunities to model and infer cellular dynamics from a systematic perspective.

Reconstructing dynamics using samples from sparsely time-resolved snapshots is an important problem in both natural sciences and machine learning. Here, we introduce a new deep learning approach for solving regularized unbalanced optimal transport (RUOT) and inferring continuous unbalanced stochastic dynamics from observed snapshots. Based on the RUOT form, our method models these dynamics without requiring prior knowledge of growth and death processes or additional information, allowing them to be learnt directly from data. Theoretically, we explore the connections between the RUOT and Schr\"odinger bridge problem and discuss the key challenges and potential solutions. The effectiveness of our method is demonstrated with a synthetic gene regulatory network. Compared with other methods, our approach accurately identifies growth and transition patterns, eliminates false transitions, and constructs the Waddington developmental landscape.

We present a novel yet simple deep learning approach, dubbed EPR-Net, for constructing the potential landscape of high-dimensional non-equilibrium steady state (NESS) systems. The key idea of our approach is to utilize the fact that the negative potential gradient is the orthogonal projection of the driving force in a weighted Hilbert space with respect to the steady-state distribution. The constructed loss function also coincides with the entropy production rate (EPR) formula in NESS theory. This approach can be extended to dealing with dimensionality reduction and state-dependent diffusion coefficients in a unified fashion. The robustness and effectiveness of the proposed approach are demonstrated by numerical studies of several high-dimensional biophysical models with multi-stability, limit cycle, or strange attractor with non-vanishing noise.

In vision-based reinforcement learning (RL) tasks, it is prevalent to assign the auxiliary task with a surrogate self-supervised loss so as to obtain more semantic representations and improve sample efficiency. However, abundant information in self-supervised auxiliary tasks has been disregarded, since the representation learning part and the decision-making part are separated. To sufficiently utilize information in the auxiliary task, we present a simple yet effective idea to employ self-supervised loss as an intrinsic reward, called Intrinsically Motivated Self-Supervised learning in Reinforcement learning (IM-SSR). We formally show that the self-supervised loss can be decomposed as exploration for novel states and robustness improvement from nuisance elimination. IM-SSR can be effortlessly plugged into any reinforcement learning with self-supervised auxiliary objectives with nearly no additional cost. Combined with IM-SSR, the previous underlying algorithms achieve salient improvements on both sample efficiency and generalization in various vision-based robotics tasks from the DeepMind Control Suite, especially when the reward signal is sparse.