Physics-informed neural networks (PINNs) have been widely used to solve partial differential equations (PDEs) in a forward and inverse manner using neural networks. However, balancing individual loss terms can be challenging, mainly when training these networks for stiff PDEs and scenarios requiring enforcement of numerous constraints. Even though statistical methods can be applied to assign relative weights to the regression loss for data, assigning relative weights to equation-based loss terms remains a formidable task. This paper proposes a method for assigning relative weights to the mean squared loss terms in the objective function used to train PINNs. Due to the presence of temporal gradients in the governing equation, the physics-informed loss can be recast using numerical integration through backward Euler discretization. The physics-uninformed and physics-informed networks should yield identical predictions when assessed at corresponding spatiotemporal positions. We refer to this consistency as “temporal consistency.” This approach introduces a unique method for training physics-informed neural networks (PINNs), redefining the loss function to allow for assigning relative weights with statistical properties of the observed data. In this work, we consider the two- and three-dimensional Navier–Stokes equations and determine the kinematic viscosity using the spatiotemporal data on the velocity and pressure fields. We consider numerical datasets to test our method. We test the sensitivity of our method to the timestep size, the number of timesteps, noise in the data, and spatial resolution. Finally, we use the velocity field obtained using particle image velocimetry experiments to generate a reference pressure field and test our framework using the velocity and pressure fields.

Blood-derived products, particularly platelet-rich plasma (PRP), have received increased attention in the past several years due to their great potential as a therapy for osteoarthritis and tendon injuries. Therefore, characterizing the mechanical properties of PRP becomes important to better understand its therapeutic efficacy. This paper aims to investigate the rheological properties of PRP in order to provide further insight into its mechanism of action. The PRP samples in our study may have higher erythrocyte count than the average PRP reported in the literature. Flow-sweep, small amplitude oscillatory shear (SAOS), large amplitude oscillatory shear (LAOS), and thixotropy tests have been performed at room and physiological temperatures to characterize the non-Newtonian properties of PRP samples. Flow-sweep tests reveal shear-thinning behavior, with higher apparent viscosity observed at a lower temperature. Rheological models such as Carreau, Casson, power-law, and Herschel–Bulkley have been fitted to flow-sweep data with the latter showing the closest agreement. Over the acceptable narrow range of frequency in SAOS tests, the loss modulus appears to be slightly larger than the storage modulus with both moduli decreasing at higher temperature. The nonlinear viscoelastic properties are furthermore quantified through the static and dynamic LAOS analyses. For a given strain and angular frequency, the former identifies strain-softening and shear-thinning, while the latter discloses a range of other transient behavior within an oscillation cycle. Lastly, peak-hold tests have revealed consistent thixotropy in PRP solutions.

Inverse problems involving time-fractional differential equations have become increasingly important for modeling systems with memory-dependent dynamics, particularly in biotransport and viscoelastic materials. Despite their potential, these problems remain challenging due to issues of stability, non-uniqueness, and limited data availability. Recent advancements in Physics-Informed Neural Networks (PINNs) offer a data-efficient framework for solving such inverse problems, yet most implementations are restricted to integer-order derivatives. In this work, we develop a PINN-based framework tailored for inverse problems involving time-fractional derivatives. We consider two representative applications: anomalous diffusion and fractional viscoelasticity. Using both synthetic and experimental datasets, we infer key physical parameters including the generalized diffusion coefficient and the fractional derivative order in the diffusion model and the relaxation parameters in a fractional Maxwell model. Our approach incorporates a customized residual loss function scaled by the standard deviation of observed data to enhance robustness. Even under 25% Gaussian noise, our method recovers model parameters with relative errors below 10%. Additionally, the framework accurately predicts relaxation moduli in porcine tissue experiments, achieving similar error margins. These results demonstrate the framework’s effectiveness in learning fractional dynamics from noisy and sparse data, paving the way for broader applications in complex biological and mechanical systems.

Lipid nanoparticle (LNP) formulation requires a mixing step. Many studies, especially from academic groups, utilize microfluidic mixers or hand mixing to prepare LNPs, while commercial-scale processes use turbulent-flow mixers. This discrepancy in mixing techniques has been underexplored, as LNPs made by different techniques may exhibit different performance, meaning bench-scale results cannot be replicated using materials manufactured at scale. Here, we isolate and interrogate the effect of primary mixing on LNP formulation. LNPs are produced from ten mixers while holding all other formulation parameters constant, to directly compare across techniques. We show that LNPs produced from the different mixers exhibit widely different physical properties and biological performance. Notably, manual pipetting common in academic practice yields particles that do not resemble those produced by turbulent-flow mixers. Findings are connected mechanistically to physicochemical characteristics arising from different flow regimes. Further establishing the relationship between mixing and LNP properties is critical.

We present a novel approach to perform passive microrheology. A method to measure the rheological properties of fluids from the Brownian motion of suspended particles. Rheological properties are found from the particles' mean square displacements (MSDs) as a function of measurement time lag. Current state-of-the-art approaches find the MSD by tracking multiple particles' trajectories. However, particle tracking approaches face many limitations, including low accuracy and high computational cost, and they are only applicable to low particle seeding densities. Here, we present a novel method, termed particle image rheometry (PIR), for estimating the particle ensemble MSD from the temporal evolution of the probability density function of the displacement as a function of measurement time lag. First, the probability density function (PDF) of the particle displacements for each time lag is found using a generalized ensemble image cross-correlation approach that eliminates the need for particle tracking. Then, PDFs are used to calculate the MSD from which the complex viscosity of the solution is measured. We evaluate the performance of PIR using synthetic datasets and show that it can achieve an error of less than 1% in passive microrheology measurements, which corresponds to a twofold lower error than existing methods. Finally, we compare the measured complex viscosity from PIR with bulk rheometry for a polymeric solution and show agreement between the two measurements.