Training machine learning tools such as neural networks require the availability of sizable data, which can be difficult for engineering and scientific applications where experiments or simulations are expensive. In this work, a novel multi-fidelity physics-constrained neural network is proposed to reduce the required amount of training data, where physical knowledge is applied to constrain neural networks, and multi-fidelity networks are constructed to improve training efficiency. A low-cost low-fidelity physics-constrained neural network is used as the baseline model, whereas a limited amount of data from a high-fidelity physics-constrained neural network is used to train a second neural network to predict the difference between the two models. The proposed framework is demonstrated with two-dimensional heat transfer, phase transition, and dendritic growth problems, which are fundamental in materials modeling. Physics is described by partial differential equations. With the same set of training data, the prediction error of physics-constrained neural network can be one order of magnitude lower than that of the classical artificial neural network without physical constraints. The accuracy of the prediction is comparable to those from direct numerical solutions of equations.

Objectives The aim of this study was to investigate serum biomarkers linked to primary Sjögren's syndrome (pSS)-associated interstitial lung disease (ILD). Methods 69 pSS patients were consecutively enrolled and evaluated via quantitative ILD scoring based on high-resolution computed tomography (HRCT). Biomarkers of interest were assessed by multiplex enzyme-linked immunosorbent assays (ELISAs). Results Among consecutively enrolled patients with pSS, the presence of pSS–ILD was 50% based on the presence of radiographically defined interstitial lung abnormalities (ILA) meeting specified criteria for mild/moderate (ILA 2) or severe (ILA 3) disease. Age, immunoglobulin M (IgM), C-reactive protein (CRP), and serum levels of eotaxin/CCL11, Krebs von den Lungen-6 (KL-6), TNFα, and TGFα were significantly higher in the combined pSS–ILD group (ILA 2 + ILA 3) than in the pSS–no-ILD and pSS–indeterminate ILD groups (ILA 0 and ILA 1, respectively) in unadjusted analyses (p < 0.05 for all variables). A binary logistic regression model revealed that disease duration and KL-6 levels were associated with the presence of pSS–ILD (p < 0.05). Complementary least absolute shrinkage and selection operator (LASSO) modeling showed that age, KL-6, and TNF-α effectively differentiated pSS–ILD (ILA 2 + ILA3) from pSS without ILD (ILA 0 + ILA 1), with an area under the curve (AUC) of 0.883 (p value < 0.0001). Conclusions Patient age, disease duration, and serum levels of both KL-6 and TNFα were the most discriminating factors associated with the presence of ILD in our pSS patients. Higher levels of CRP, IgM, eotaxin, TGFα, and TNFα should also prompt the search for occult as well as clinically evident lung involvement based on statistically significant univariate associations with pSS–ILD. Clinical trial registration None.

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