Physics-informed machine learning of the Lagrangian dynamics of velocity gradient tensor

Reduced models describing the Lagrangian dynamics of the velocity gradient tensor (VGT) in homogeneous isotropic turbulence (HIT) are developed under the physics-informed machine learning (PIML) framework. We consider the VGT at both Kolmogorov scale and coarse-grained scale within the inertial range of HIT. Building reduced models requires resolving the pressure Hessian and subfilter contributions, which is accomplished by constructing them using the integrity bases and invariants of the VGT. The developed models can be expressed using the extended tensor basis neural network (TBNN) introduced by Ling et al. [J. Fluid Mech. 807, 155 (2016)]. Physical constraints, such as Galilean invariance, rotational invariance, and incompressibility condition, are thus embedded in the models explicitly. Our PIML models are trained on the Lagrangian data from a high-Reynolds number direct numerical simulation (DNS). To validate the results, we perform a comprehensive out-of-sample test. We observe that the PIML model provides an improved representation for the magnitude and orientation of the small-scale pressure Hessian contributions. Statistics of the flow, as indicated by the joint PDF of second and third invariants of the VGT, show good agreement with the “ground-truth” DNS data. A number of other important features describing the structure of HIT are reproduced by the model successfully. We have also identified challenges in modeling inertial range dynamics, which indicates that a richer modeling strategy is required. This helps us identify important directions for future research, in particular towards including inertial range geometry into the TBNN.

Figure 1

Architecture of the extended tensor basis neural network used in this work, which is based on the TBNN structure used in Ling et al. [35] and extended to include skew-symmetric tensor bases.

Figure 2

The decay of both the training and testing loss for the (a) PIML model and (b) PB model.

Figure 3

The testing error of the incompressibility constraint and rotational invariance of the (a) PIML and (b) PB models.

Figure 4

PDFs of the angle (in radians) between the predicted $e_1$ eigenvector and DNS data. The PDF peaks at 0 when the predicted $e_1$ eigenvector is aligned with the DNS data and it peaks at $\pi /2$ in the case of misalignment.

Figure 5

PDFs of the coefficients of the tensor bases from the PIML model, ML model A, Gaussian closure [29], and Ref. [31].

Figure 6

Contributions of the different tensor bases to the prediction of nonlocal pressure Hessian for the PIML model, ML model A, Gaussian closure [29], and Ref. [31].

Figure 7

Lagrangian dynamics projected onto the $Q\text{−}R$ phase plane: (a) restricted Euler term, (b) pressure Hessian term, (c) viscous term, and (d) sum of all the terms. Zero discriminant line, $D=27R^2-4Q^3=0$, descriptive of the RE dynamics [19], is shown in blue.

Figure 8

Comparison of pressure Hessian contributions on the $Q\text{−}R$ phase plane.

Figure 9

Comparison of Lagrangian dynamics between (a) DNS and (b) PIML model on the $Q\text{−}R$ phase plane.

Figure 10

$Q\text{−}R$ plane PDF. (a) Random initialization (Gaussian, traceless VGT), (b) PIML SODE prediction, and (c) DNS.

Figure 11

Standardized PDFs of the longitudinal component: (a) $A_{11}$ and transverse component; (b): $A_{12}$ of the VGT.

Figure 12

Standardized PDFs of the PIML model error of the longitudinal $H_{11}$ and transverse $H_{12}$ components of pressure Hessian.

Figure 13

CMV of the PIML model prediction on the $Q\text{−}R$ phase plane. Both the mean (black arrows) and skewness (blue arrows) are shown with zoomed-in view of the third and fourth quadrants.

Figure 14

CMV of the coarse-grained VGT dynamics with filter scale selected to be, $l_{\mathrm{\Delta }}=\pi /8$. (a) restricted Euler, (b) pressure Hessian, (c) viscous term, and (d) subfilter contribution.

Figure 15

PDF of the angle (radian) between the predicted $e_1$ eigenvector and DNS data for various filter sizes.

Figure 16

PDF of the error of PIML model prediction for various filter sizes. The curves are renormalized to avoid clustering.

Figure 17

Comparison of subfilter CMV between the PIML model prediction and DNS. PIML model prediction is shown as red and DNS data are shown as black. The chosen coarse-graining size is $\pi /8$.