We study recovering fluid density and velocity from sparse multiview videos.

The following is an extract from our Lost in Space-Time newsletter. Each month, we hand over the keyboard to a physicist or mathematician to tell you about fascinating ideas from their corner of the universe. You can sign up for Lost in Space-Time here. The Navier-Stokes equations have been used to model the flow of fluids for almost 200 years – but we still don't really understand them. This can often feel a little odd, especially as we rely on these equations every day to help build rockets, design drugs and understand climate change. But here is where you have to think like a mathematician.

Gaussian process regression techniques have been used in fluid mechanics for the reconstruction of flow fields from a reduction-of-dimension perspective. A main ingredient in this setting is the construction of adapted covariance functions, or kernels, to obtain such estimates. In this paper, we derive physics-informed kernels for simulating two-dimensional velocity fields of an incompressible (divergence-free) flow around aerodynamic profiles. These kernels allow to define Gaussian process priors satisfying the incompressibility condition and the prescribed boundary conditions along the profile in a continuous manner. Such physical and boundary constraints can be applied to any pre-defined scalar kernel in the proposed methodology, which is very general and can be implemented with high flexibility for a broad range of engineering applications. Its relevance and performances are illustrated by numerical simulations of flows around a cylinder and a NACA 0412 airfoil profile, for which no observation at the boundary is needed at all.

We present a robust neural watermarking framework for scientific data integrity, targeting high-dimensional fields common in climate modeling and fluid simulations. Using a convolutional autoencoder, binary messages are invisibly embedded into structured data such as temperature, vorticity, and geopotential. Our method ensures watermark persistence under lossy transformations - including noise injection, cropping, and compression - while maintaining near-original fidelity (sub-1\% MSE). Compared to classical singular value decomposition (SVD)-based watermarking, our approach achieves $>$98\% bit accuracy and visually indistinguishable reconstructions across ERA5 and Navier-Stokes datasets. This system offers a scalable, model-compatible tool for data provenance, auditability, and traceability in high-performance scientific workflows, and contributes to the broader goal of securing AI systems through verifiable, physics-aware watermarking. We evaluate on physically grounded scientific datasets as a representative stress-test; the framework extends naturally to other structured domains such as satellite imagery and autonomous-vehicle perception streams.

This work presents a data-driven solution to accurately predict parameterized nonlinear fluid dynamical systems using a dynamics-generator conditional GAN (Dyn-cGAN) as a surrogate model. The Dyn-cGAN includes a dynamics block within a modified conditional GAN, enabling the simultaneous identification of temporal dynamics and their dependence on system parameters. The learned Dyn-cGAN model takes into account the system parameters to predict the flow fields of the system accurately. We evaluate the effectiveness and limitations of the developed Dyn-cGAN through numerical studies of various parameterized nonlinear fluid dynamical systems, including flow over a cylinder and a 2-D cavity problem, with different Reynolds numbers. Furthermore, we examine how Reynolds number affects the accuracy of the predictions for both case studies. Additionally, we investigate the impact of the number of time steps involved in the process of dynamics block training on the accuracy of predictions, and we find that an optimal value exists based on errors and mutual information relative to the ground truth.

Microswimmers are sub-millimeter swimming microrobots that show potential as a platform for controllable locomotion in applications including targeted cargo delivery and minimally invasive surgery. To be viable for these target applications, microswimmers will eventually need to be able to navigate in environments with dynamic fluid flows and forces. Experimental studies with microswimmers towards this goal are currently rare because of the difficulty isolating intentional microswimmer motion from environment-induced motion. In this work, we present a method for measuring microswimmer locomotion within a complex flow environment using fiducial microspheres. By tracking the particle motion of ferromagnetic and non-magnetic polystyrene fiducial microspheres, we capture the effect of fluid flow and field gradients on microswimmer trajectories. We then determine the field-driven translation of these microswimmers relative to fluid flow and demonstrate the effectiveness of this method by illustrating the motion of multiple microswimmers through different flows.

We present a generative AI algorithm for addressing the challenging task of fast, accurate and robust statistical computation of three-dimensional turbulent fluid flows. Our algorithm, termed as GenCFD, is based on a conditional score-based diffusion model. Through extensive numerical experimentation with both incompressible and compressible fluid flows, we demonstrate that GenCFD provides very accurate approximation of statistical quantities of interest such as mean, variance, point pdfs, higher-order moments, while also generating high quality realistic samples of turbulent fluid flows and ensuring excellent spectral resolution. In contrast, ensembles of operator learning baselines which are trained to minimize mean (absolute) square errors regress to the mean flow. We present rigorous theoretical results uncovering the surprising mechanisms through which diffusion models accurately generate fluid flows. These mechanisms are illustrated with solvable toy models that exhibit the relevant features of turbulent fluid flows while being amenable to explicit analytical formulas.

In our previous paper [N. Tsutsumi, K. Nakai and Y. Saiki, Chaos 32, 091101 (2022)], we proposed a method for constructing a system of differential equations of chaotic behavior from only observable deterministic time series, which we call the radial function-based regression (RfR) method. However, when the targeted variable's behavior is rather complex, the direct application of the RfR method does not function well. In this study, we propose a novel method of modeling such dynamics, including the high-frequency intermittent behavior of a fluid flow, by considering another variable (base variable) showing relatively simple, less intermittent behavior. We construct an autonomous joint model composed of two parts: the first is an autonomous system of a base variable, and the other concerns the targeted variable being affected by a term involving the base variable to demonstrate complex dynamics. The constructed joint model succeeded in not only inferring a short trajectory but also reconstructing chaotic sets and statistical properties obtained from a long trajectory such as the density distributions of the actual dynamics.

Scientific machine learning (SciML) methods such as physics-informed neural networks (PINNs) are used to estimate parameters of interest from governing equations and small quantities of data. However, there has been little work in assessing how well PINNs perform for inverse problems across wide ranges of governing equations across the mathematical sciences. We present a new and challenging benchmark problem for inverse PINNs based on a parametric sweep of the 2D Burgers' equation with rotational flow. We show that a novel strategy that alternates between first- and second-order optimization proves superior to typical first-order strategies for estimating parameters. In addition, we propose a novel data-driven method to characterize PINN effectiveness in the inverse setting. PINNs' physics-informed regularization enables them to leverage small quantities of data more efficiently than the data-driven baseline. However, both PINNs and the baseline can fail to recover parameters for highly inviscid flows, motivating the need for further development of PINN methods.