Stochastic partial differential equations (SPDEs) are significant tools for modeling dynamics in many areas including atmospheric sciences and physics. Neural Operators, generations of neural networks with capability of learning maps between infinite-dimensional spaces, are strong tools for solving parametric PDEs. However, they lack the ability to modeling SPDEs which usually have poor regularity 1 due to the driving noise. As the theory of regularity structure has achieved great successes in analyzing SPDEs and provides the concept model feature vectors that well-approximate SPDEs' solutions, we propose the Neural Operator with Regularity Structure (NORS) which incorporates the feature vectors for modeling dynamics driven by SPDEs. We conduct experiments on various of SPDEs including the dynamic Φ 4 1 model and the 2d stochastic Navier-Stokes equation, and the results demonstrate that the NORS is resolution-invariant, efficient, and achieves one order of magnitude lower error with a modest amount of data.
Purpose Community-acquired pneumonia (CAP) is one of the most frequently encountered infectious diseases worldwide. Few studies have explored the microbial composition of the lower respiratory tract (LRT) and host metabolites of CAP. We analyzed the microbial composition of the LRT and levels of host metabolites to explore new biomarkers for CAP. Patients and Methods Bronchoalveolar lavage fluid (BALF) was collected from 28 CAP patients and 20 healthy individuals. Following centrifugation, BALF pellets were used for amplicon sequencing of a variable region of the bacterial 16S rDNA gene to characterize the microbial composition. Non-targeted metabolomics was used to detect host’s metabolites in the supernatant. Results Compared with healthy individuals, the bacterial alpha diversity in the LRT of CAP patients was significantly lower in CAP patients (p<0.05). On the bacterial genus level, over 20 genera were detected with lower relative abundance (p<0.05), while the relative abundance of Ruminiclostridium -6 was significantly higher in CAP patients. The levels of the host metabolites dimethyldisulfide, choline, pyrimidine, oleic acid and N-acetyl-neuraminic acid were all increased in BALF of CAP patients (p<0.05), while concentrations of lysophosphatidylcholines (LPC (12:0/0:0)) and phosphatidic acid (PA (20:4/2:0)) were decreased (p<0.05). Furthermore, the relative abundance of Parvimonas, Treponema -2, Moraxella, Aggregatibacter, Filifactor, Fusobacterium, Lautropia and Neisseria negatively correlated with concentrations of oleic acid (p<0.05). A negative correlation between the relative abundance of Treponema -2, Moraxella, Filifactor, Fusobacterium and dimethyldisulfide concentrations was also observed (p<0.05). In contrast, the relative abundance of Treponema -2, Moraxella, Filifactor , and Fusobacterium was found to be positively associated with concentrations of LPC (12:0/0:0) and PA (20:4/2:0) (p<0.05). Conclusion The composition of the LRT microbiome differed between healthy individuals and CAP patients. We propose that some respiratory microbial components and host metabolites are potentially novel diagnostic markers of CAP.
We present the deep random vortex network (DRVN), a novel physics-informed framework for simulating and inferring the fluid dynamics governed by the incompressible Navier--Stokes equations. Unlike the existing physics-informed neural network (PINN), which embeds physical and geometry information through the residual of equations and boundary data, DRVN automatically embeds this information into neural networks through neural random vortex dynamics equivalent to the Navier--Stokes equation. Specifically, the neural random vortex dynamics motivates a Monte Carlo-based loss function for training neural networks, which avoids the calculation of derivatives through auto-differentiation. Therefore, DRVN can efficiently solve Navier--Stokes equations with non-differentiable initial conditions and fractional operators. Furthermore, DRVN naturally embeds the boundary conditions into the kernel function of the neural random vortex dynamics and thus, does not need additional data to obtain boundary information. We conduct experiments on forward and inverse problems with incompressible Navier--Stokes equations. The proposed method achieves accurate results when simulating and when inferring Navier--Stokes equations. For situations that include singular initial conditions and agnostic boundary data, DRVN significantly outperforms the existing PINN method. Furthermore, compared with the conventional adjoint method when solving inverse problems, DRVN achieves a 2 orders of magnitude improvement for the training time with significantly precise estimates.
Stochastic partial differential equations (SPDEs) are crucial for modelling dynamics with randomness in many areas including economics, physics, and atmospheric sciences. Recently, using deep learning approaches to learn the PDE solution for accelerating PDE simulation becomes increasingly popular. However, SPDEs have two unique properties that require new design on the models. First, the model to approximate the solution of SPDE should be generalizable over both initial conditions and the random sampled forcing term. Second, the random forcing terms usually have poor regularity whose statistics may diverge (e.g., the space-time white noise). To deal with the problems, in this work, we design a deep neural network called \emph{Deep Latent Regularity Net} (DLR-Net). DLR-Net includes a regularity feature block as the main component, which maps the initial condition and the random forcing term to a set of regularity features. The processing of regularity features is inspired by regularity structure theory and the features provably compose a set of basis to represent the SPDE solution. The regularity features are then fed into a small backbone neural operator to get the output. We conduct experiments on various SPDEs including the dynamic $\Phi^4_1$ model and the stochastic 2D Navier-Stokes equation to predict their solutions, and the results demonstrate that the proposed DLR-Net can achieve SOTA accuracy compared with the baselines. Moreover, the inference time is over 20 times faster than the traditional numerical solver and is comparable with the baseline deep learning models.
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