Yen Ting Lin

Afroza Shirin, Isaac Klickstein, Song Feng et al.

The effects of molecularly targeted drug perturbations on cellular activities and fates are difficult to predict using intuition alone because of the complex behaviors of cellular regulatory networks. An approach to overcoming this problem is to develop mathematical models for predicting drug effects. Such an approach beckons for co-development of computational methods for extracting insights useful for guiding therapy selection and optimizing drug scheduling. Here, we present and evaluate a generalizable strategy for identifying drug dosing schedules that minimize the amount of drug needed to achieve sustained suppression or elevation of an important cellular activity/process, the recycling of cytoplasmic contents through (macro)autophagy. Therapeutic targeting of autophagy is currently being evaluated in diverse clinical trials but without the benefit of a control engineering perspective. Using a nonlinear ordinary differential equation (ODE) model that accounts for activating and inhibiting influences among protein and lipid kinases that regulate autophagy (MTORC1, ULK1, AMPK and VPS34) and methods guaranteed to find locally optimal control strategies, we find optimal drug dosing schedules (open-loop controllers) for each of six classes of drugs and drug pairs. Our approach is generalizable to designing monotherapy and multi therapy drug schedules that affect different cell signaling networks of interest.

Javier E. Santos, Yen Ting Lin

The aim of this short note is to show that Denoising Diffusion Probabilistic Model DDPM, a non-homogeneous discrete-time Markov process, can be represented by a time-homogeneous continuous-time Markov process observed at non-uniformly sampled discrete times. Surprisingly, this continuous-time Markov process is the well-known and well-studied Ornstein-Ohlenbeck (OU) process, which was developed in 1930's for studying Brownian particles in Harmonic potentials. We establish the formal equivalence between DDPM and the OU process using its analytical solution. We further demonstrate that the design problem of the noise scheduler for non-homogeneous DDPM is equivalent to designing observation times for the OU process. We present several heuristic designs for observation times based on principled quantities such as auto-variance and Fisher Information and connect them to ad hoc noise schedules for DDPM. Interestingly, we show that the Fisher-Information-motivated schedule corresponds exactly the cosine schedule, which was developed without any theoretical foundation but is the current state-of-the-art noise schedule.

Jinpai Zhao, Nishant Panda, Yen Ting Lin et al.

We introduce HyCOP, a modular framework that learns parametric PDE solution operators by composing simple modules (advection, diffusion, learned closures, boundary handling) in a query-conditioned way. Rather than learning a monolithic map, HyCOP learns a policy over short programs - which module to apply and for how long - conditioned on regime features and state statistics. Modules may be numerical sub-solvers or learned components, enabling hybrid surrogates evaluated at arbitrary query times without autoregressive rollout. Across diverse PDE benchmarks, HyCOP produces interpretable programs, delivers order-of-magnitude OOD improvements over monolithic neural operators, and supports modular transfer through dictionary updates (e.g., boundary swaps, residual enrichment). Our theory characterizes expressivity and gives an error decomposition that separates composition error from module error and doubles as a process-level diagnostic.

Yifeng Tian, Nishant Panda, Yen Ting Lin

We present the Liouville Flow Importance Sampler (LFIS), an innovative flow-based model for generating samples from unnormalized density functions. LFIS learns a time-dependent velocity field that deterministically transports samples from a simple initial distribution to a complex target distribution, guided by a prescribed path of annealed distributions. The training of LFIS utilizes a unique method that enforces the structure of a derived partial differential equation to neural networks modeling velocity fields. By considering the neural velocity field as an importance sampler, sample weights can be computed through accumulating errors along the sample trajectories driven by neural velocity fields, ensuring unbiased and consistent estimation of statistical quantities. We demonstrate the effectiveness of LFIS through its application to a range of benchmark problems, on many of which LFIS achieved state-of-the-art performance.

An B. Vuong, Michael T. McCann, Javier E. Santos et al.

Diffusion models are commonly interpreted as learning the score function, i.e., the gradient of the log-density of noisy data. However, this assumption implies that the target of learning is a conservative vector field, which is not enforced by the neural network architectures used in practice. We present numerical evidence that trained diffusion networks violate both integral and differential constraints required of true score functions, demonstrating that the learned vector fields are not conservative. Despite this, the models perform remarkably well as generative mechanisms. To explain this apparent paradox, we advocate a new theoretical perspective: diffusion training is better understood as flow matching to the velocity field of a Wasserstein Gradient Flow (WGF), rather than as score learning for a reverse-time stochastic differential equation. Under this view, the "probability flow" arises naturally from the WGF framework, eliminating the need to invoke reverse-time SDE theory and clarifying why generative sampling remains successful even when the neural vector field is not a true score. We further show that non-conservative errors from neural approximation do not necessarily harm density transport. Our results advocate for adopting the WGF perspective as a principled, elegant, and theoretically grounded framework for understanding diffusion generative models.

Javier E. Santos, Agnese Marcato, Roman Colman et al.

Generative diffusion models have achieved remarkable success in producing high-quality images. However, these models typically operate in continuous intensity spaces, diffusing independently across pixels and color channels. As a result, they are fundamentally ill-suited for applications involving inherently discrete quantities-such as particle counts or material units-that are constrained by strict conservation laws like mass conservation, limiting their applicability in scientific workflows. To address this limitation, we propose Discrete Spatial Diffusion (DSD), a framework based on a continuous-time, discrete-state jump stochastic process that operates directly in discrete spatial domains while strictly preserving particle counts in both forward and reverse diffusion processes. By using spatial diffusion to achieve particle conservation, we introduce stochasticity naturally through a discrete formulation. We demonstrate the expressive flexibility of DSD by performing image synthesis, class conditioning, and image inpainting across standard image benchmarks, while exactly conditioning total image intensity. We validate DSD on two challenging scientific applications: porous rock microstructures and lithium-ion battery electrodes, demonstrating its ability to generate structurally realistic samples under strict mass conservation constraints, with quantitative evaluation using state-of-the-art metrics for transport and electrochemical performance.

Xander de Wit, Alessandro Gabbana, Michael Woodward et al.

The dynamics of Lagrangian particles in turbulence play a crucial role in mixing, transport, and dispersion processes in complex flows. Their trajectories exhibit highly non-trivial statistical behavior, motivating the development of surrogate models that can reproduce these trajectories without incurring the high computational cost of direct numerical simulations of the full Eulerian field. This task is particularly challenging because reduced-order models typically lack access to the full set of interactions with the underlying turbulent field. Novel data-driven machine learning techniques can be very powerful in capturing and reproducing complex statistics of the reduced-order/surrogate dynamics. In this work, we show how one can learn a surrogate dynamical system that is able to evolve a turbulent Lagrangian trajectory in a way that is point-wise accurate for short-time predictions (with respect to Kolmogorov time) and stable and statistically accurate at long times. This approach is based on the Mori--Zwanzig formalism, which prescribes a mathematical decomposition of the full dynamical system into resolved dynamics that depend on the current state and the past history of a reduced set of observables and the unresolved orthogonal dynamics due to unresolved degrees of freedom of the initial state. We show how by training this reduced order model on a point-wise error metric on short time-prediction, we are able to correctly learn the dynamics of the Lagrangian turbulence, such that also the long-time statistical behavior is stably recovered at test time. This opens up a range of new applications, for example, for the control of active Lagrangian agents in turbulence.

Javier E Santos, Zachary R. Fox, Nicholas Lubbers et al.

Typical generative diffusion models rely on a Gaussian diffusion process for training the backward transformations, which can then be used to generate samples from Gaussian noise. However, real world data often takes place in discrete-state spaces, including many scientific applications. Here, we develop a theoretical formulation for arbitrary discrete-state Markov processes in the forward diffusion process using exact (as opposed to variational) analysis. We relate the theory to the existing continuous-state Gaussian diffusion as well as other approaches to discrete diffusion, and identify the corresponding reverse-time stochastic process and score function in the continuous-time setting, and the reverse-time mapping in the discrete-time setting. As an example of this framework, we introduce ``Blackout Diffusion'', which learns to produce samples from an empty image instead of from noise. Numerical experiments on the CIFAR-10, Binarized MNIST, and CelebA datasets confirm the feasibility of our approach. Generalizing from specific (Gaussian) forward processes to discrete-state processes without a variational approximation sheds light on how to interpret diffusion models, which we discuss.

Xiaoyu Ma, Sylvain Sardy, Nick Hengartner et al.

To fit sparse linear associations, a LASSO sparsity inducing penalty with a single hyperparameter provably allows to recover the important features (needles) with high probability in certain regimes even if the sample size is smaller than the dimension of the input vector (haystack). More recently learners known as artificial neural networks (ANN) have shown great successes in many machine learning tasks, in particular fitting nonlinear associations. Small learning rate, stochastic gradient descent algorithm and large training set help to cope with the explosion in the number of parameters present in deep neural networks. Yet few ANN learners have been developed and studied to find needles in nonlinear haystacks. Driven by a single hyperparameter, our ANN learner, like for sparse linear associations, exhibits a phase transition in the probability of retrieving the needles, which we do not observe with other ANN learners. To select our penalty parameter, we generalize the universal threshold of Donoho and Johnstone (1994) which is a better rule than the conservative (too many false detections) and expensive cross-validation. In the spirit of simulated annealing, we propose a warm-start sparsity inducing algorithm to solve the high-dimensional, non-convex and non-differentiable optimization problem. We perform precise Monte Carlo simulations to show the effectiveness of our approach.

Sylvain Sardy, Nicolas W Hengartner, Nikolai Bonenko et al.

Using a sparsity inducing penalty in artificial neural networks (ANNs) avoids over-fitting, especially in situations where noise is high and the training set is small in comparison to the number of features. For linear models, such an approach provably also recovers the important features with high probability in regimes for a well-chosen penalty parameter. The typical way of setting the penalty parameter is by splitting the data set and performing the cross-validation, which is (1) computationally expensive and (2) not desirable when the data set is already small to be further split (for example, whole-genome sequence data). In this study, we establish the theoretical foundation to select the penalty parameter without cross-validation based on bounding with a high probability the infinite norm of the gradient of the loss function at zero under the zero-feature assumption. Our approach is a generalization of the universal threshold of Donoho and Johnstone (1994) to nonlinear ANN learning. We perform a set of comprehensive Monte Carlo simulations on a simple model, and the numerical results show the effectiveness of the proposed approach.