Understanding the principles that govern dynamical systems is a central challenge across many scientific domains, including biology and ecology. Incomplete knowledge of nonlinear interactions and stochastic effects often renders bottom-up modeling approaches ineffective, motivating the development of methods that can discover governing equations directly from data. In such contexts, parametric models often struggle without strong prior knowledge, especially when estimating intrinsic noise. Nonetheless, incorporating stochastic effects is often essential for understanding the dynamic behavior of complex systems such as gene regulatory networks and signaling pathways. To address these challenges, we introduce Trine (Three-phase Regression for INtrinsic noisE), a nonparametric, kernel-based framework that infers state-dependent intrinsic noise from time-series data. Trine features a three-stage algorithm that com- bines analytically solvable subproblems with a structured kernel architecture that captures both abrupt noise-driven fluctuations and smooth, state-dependent changes in variance. We validate Trine on biological and ecological systems, demonstrating its ability to uncover hidden dynamics without relying on predefined parametric assumptions. Across several benchmark problems, Trine achieves performance comparable to that of an oracle. Biologically, this oracle can be viewed as an idealized observer capable of directly tracking the random fluctuations in molecular concentrations or reaction events within a cell. The Trine framework thus opens new avenues for understanding how intrinsic noise affects the behavior of complex systems.

A variety of forces shape neural representations in the brain.

We thank the reviewers (R1, R2, R3) for their detailed feedback. Key concerns are briefly addressed below. Kolen & Pollack, 1994), so symmetry is an emergent property of the network. Prior manipulations were made throughout learning, not only at test time. "It is not clear in what way intrinsic noise in the recurrent dynamics ... allows us to derive closed-form probabilistic ": when we say that noise is necessary for our learning rule, we mean Intrinsic noise is what allows for this probabilistic description.

Inferring dynamical models from data continues to be a significant challenge in computational biology, especially given the stochastic nature of many biological processes. We explore a common scenario in omics, where statistically independent cross-sectional samples are available at a few time points, and the goal is to infer the underlying diffusion process that generated the data. Existing inference approaches often simplify or ignore noise intrinsic to the system, compromising accuracy for the sake of optimization ease. We circumvent this compromise by inferring the phase-space probability flow that shares the same time-dependent marginal distributions as the underlying stochastic process. Our approach, probability flow inference (PFI), disentangles force from intrinsic stochasticity while retaining the algorithmic ease of ODE inference. Analytically, we prove that for Ornstein-Uhlenbeck processes the regularized PFI formalism yields a unique solution in the limit of well-sampled distributions. In practical applications, we show that PFI enables accurate parameter and force estimation in high-dimensional stochastic reaction networks, and that it allows inference of cell differentiation dynamics with molecular noise, outperforming state-of-the-art approaches.

Multi-modal fusion is crucial in medical data research, enabling a comprehensive understanding of diseases and improving diagnostic performance by combining diverse modalities. However, multi-modal fusion faces challenges, including capturing interactions between modalities, addressing missing modalities, handling erroneous modal information, and ensuring interpretability. Many existing researchers tend to design different solutions for these problems, often overlooking the commonalities among them. This paper proposes a novel multi-modal fusion framework that achieves adaptive adjustment over the weights of each modality by introducing the Modal-Domain Attention (MDA). It aims to facilitate the fusion of multi-modal information while allowing for the inclusion of missing modalities or intrinsic noise, thereby enhancing the representation of multi-modal data. We provide visualizations of accuracy changes and MDA weights by observing the process of modal fusion, offering a comprehensive analysis of its interpretability. Extensive experiments on various gastrointestinal disease benchmarks, the proposed MDA maintains high accuracy even in the presence of missing modalities and intrinsic noise. One thing worth mentioning is that the visualization of MDA is highly consistent with the conclusions of existing clinical studies on the dependence of different diseases on various modalities. Code and dataset will be made available.

Engineering and applied science rely on computational experiments to rigorously study physical systems. The mathematical models used to probe these systems are highly complex, and sampling-intensive studies often require prohibitively many simulations for acceptable accuracy. Surrogate models provide a means of circumventing the high computational expense of sampling such complex models. In particular, polynomial chaos expansions (PCEs) have been successfully used for uncertainty quantification studies of deterministic models where the dominant source of uncertainty is parametric. We discuss an extension to conventional PCE surrogate modeling to enable surrogate construction for stochastic computational models that have intrinsic noise in addition to parametric uncertainty. We develop a PCE surrogate on a joint space of intrinsic and parametric uncertainty, enabled by Rosenblatt transformations, and then extend the construction to random field data via the Karhunen-Loeve expansion. We then take advantage of closed-form solutions for computing PCE Sobol indices to perform a global sensitivity analysis of the model which quantifies the intrinsic noise contribution to the overall model output variance. Additionally, the resulting joint PCE is generative in the sense that it allows generating random realizations at any input parameter setting that are statistically approximately equivalent to realizations from the underlying stochastic model. The method is demonstrated on a chemical catalysis example model.

The review analyzes the fundamental principles which Artificial Intelligence should be based on in order to imitate the realistic process of taking decisions by humans experiencing emotions. Two approaches are compared, one based on quantum theory and the other employing classical terms. Both these approaches have a number of similarities, being principally probabilistic. The analogies between quantum measurements under intrinsic noise and affective decision making are elucidated. It is shown that cognitive processes have many features that are formally similar to quantum measurements. This, however, in no way means that for the imitation of human decision making Affective Artificial Intelligence has necessarily to rely on the functioning of quantum systems. Appreciating the common features between quantum measurements and decision making helps for the formulation of an axiomatic approach employing only classical notions. Artificial Intelligence, following this approach, operates similarly to humans, by taking into account the utility of the considered alternatives as well as their emotional attractiveness. Affective Artificial Intelligence, whose operation takes account of the cognition-emotion duality, avoids numerous behavioural paradoxes of traditional decision making. A society of intelligent agents, interacting through the repeated multistep exchange of information, forms a network accomplishing dynamic decision making. The considered intelligent networks can characterize the operation of either a human society of affective decision makers, or the brain composed of neurons, or a typical probabilistic network of an artificial intelligence.

Stephanie L. Hyland Microsoft Research Shruti Tople Microsoft Research Abstract --Protecting the privacy of training data is important for the safe deployment of machine learning models. Private learning algorithms have been proposed that ensure strong differential-privacy (DP) guarantees. However, the additional noise required for such protection comes at the cost of reduced model utility. Meanwhile, the stochastic gradient descent (SGD) method -- the most common optimization algorithm for neural networks -- contains intrinsic randomness which has not been leveraged for privacy. Arguing that SGD guarantees intrinsic privacy, we investigate the extent to which this privacy can be quantified and used to improve the utility of privately learned models. In effect, we ask the question; "If SGD were a differentially-private mechanism, how good would it be?" In this work, we take the first step towards analysing the intrinsic privacy properties of SGD. Our primary contribution is a large-scale empirical analysis of SGD on both convex and non-convex objectives. T o this end, we evaluate the inherent variability due to the stochasticity in SGD on 3 different datasets and calculate the null values due to the intrinsic noise. First, we show that the variability in model parameters due to the random sampling almost always exceeds that due to changes in the data. We observe that SGD provides intrinsic null values of 7. 8, 6 .9 Next, we propose a method to augment the intrinsic noise of SGD with additional noise to achieve the desired null. Our augmented SGD outputs model that outperform existing approaches with the same privacy guarantee, thus closing the gap to noiseless utility between 0 . Finally, we show that the existing theoretical bound on the sensitivity of SGD is not tight. By estimating the tightest bound empirically, we achieve near-noiseless performance at null 1, closing the utility gap to the noiseless model between 3 . Our experiments provide concrete evidence that changing the seed in SGD is likely to have a far greater impact on the resulting model than including or excluding any given training example. By properly accounting for this intrinsic randomness, higher utility can be achieved without sacrificing further privacy. With these results, we hope to inspire the research community to further explore and characterise the randomness in SGD, its impact on privacy, and the parallels with generalisation in machine learning. I NTRODUCTION Respecting the privacy of users contributing their data to train machine learning models is important.