Neural Velocity Flow Network

• Neural velocity flow networks are deep learning models that reconstruct spatiotemporal velocity fields in turbulent flows while preserving physical symmetries.
• The network employs a tensor basis architecture with 11 hidden layers and invariant features to accurately model the pressure–Hessian tensor from local velocity gradients.
• Element-wise normalization and constant-coefficient representations enhance eigenvector alignment statistics, offering practical improvements in turbulence closure modeling.

A neural velocity flow network is a class of deep learning model specifically designed to infer, reconstruct, or predict spatiotemporal velocity fields in complex, nonlinear fluid flows, often underpinned by physical constraints or symmetries. Such networks have been employed to extract closures (e.g., pressure–Hessian) from local gradients, reconstruct hidden variables from sparse or indirect measurements, or embed governing physics into their mapping structures, thereby serving as essential tools in turbulence modeling, experimental diagnostics, and data-driven closure development.

1. Tensor Basis Neural Network Architecture

The tensor basis neural network (TBNN) embeds fundamental physical principles—specifically rotational invariance and the tensor integrity basis—within its architecture. The pressure–Hessian's deviatoric part, $P_{tf}$, is modeled as a sum over ten integrity basis tensors $T^i$ formed from the local symmetric (strain-rate $S$) and antisymmetric (rotation-rate $R$) parts of the velocity gradient tensor $A$:

$P_{tf}^{(\mathrm{TBNN})} = \sum_{i=1}^{10} C^i T^i$

where:

• $T^1 = S$
• $T^2 = SR - RS$
• $T^3 = S^2 - \frac{1}{3} I \operatorname{tr}(S^2)$
• $T^4 = R^2 - \frac{1}{3} I \operatorname{tr}(R^2)$
• $T^5 = RS^2 - S^2R$
• $$T^6 = R^2 S + S R^2 - \frac{2}{3} I \operatorname{tr}(S R^2)$$
• $T^7 = RSR^2 - R^2SR$
• $T^8 = S R S^2 - S^2 R S$
• $$T^9 = R^2 S^2 + S^2 R^2 - \frac{2}{3} I \operatorname{tr}(S^2 R^2)$$
• $T^{10} = RS^2 R^2 - R^2 S^2 R$

The scalar coefficients $C^i$ output by the neural network depend on five invariants: $$\{\lambda^i\} = \{\operatorname{tr}(S^2), \operatorname{tr}(R^2), \operatorname{tr}(S^3), \operatorname{tr}(R^2 S), \operatorname{tr}(R^2 S^2)\}$$, which serve as input features. The Cayley–Hamilton theorem and tensor basis formalism guarantee that the mapping preserves all tensor symmetries and rotational invariance. The network is constructed with 11 hidden layers (with neuron configuration such as 50, 150, ..., 100) and RELU activations, and is trained to minimize the Frobenius-norm discrepancy between network and DNS pressure–Hessian tensors.

A novel modification involves element-wise normalization of the tensor basis $T^i$ (scaling to [0, 1]), improving the reproduction of physical eigenvector alignments at the expense of losing strict tensor invariance.

2. Training Regime and Data Considerations

The TBNN is trained on high-resolution DNS data from the John Hopkins Turbulence Database (JHTD), at Taylor-scale Reynolds number $Re_\lambda = 433$. The dataset comprises 262,144 samples of velocity gradient tensor $A$ and pressure–Hessian tensors, divided into 236,544 for training and 25,600 for cross-validation. Each batch contains 256 samples, with weights initialized via the Glorot normal method and training optimized using RMSprop (learning rate $1.0 \times 10^{-6}$) until cost stagnation.

The velocity gradient tensor is non-dimensionalized by the mean Frobenius norm across the dataset, but initially, invariants and basis tensors are unnormalized. Later experiments demonstrate that element-wise normalization of $T^i$ is crucial for writing pressure–Hessian as a physically accurate function of local gradients.

The trained model is further evaluated on: (i) an additional JHTD dataset at $Re=433$, (ii) a UP Madrid dataset at $Re=315$, and (iii) channel flow data from UT Texas and JHTD at a friction Reynolds number of 1000, thereby probing transferability across flow regimes.

3. Model Evaluation: Alignment Statistics and Physical Metrics

Performance is assessed not only via relative Frobenius-norm error ($\approx 0.65$ for TBNN vs $0.78$ for RFDM) but, crucially, through statistics of eigenvector alignment between modeled pressure–Hessian and strain–rate tensor. Specifically, the probability density functions (PDFs) of the cosine of the angle between the leading eigenvector of $P_{tf}$ and $S$ are computed. The RFDM produces eigenvector alignments that are almost parallel or perpendicular (a nonphysical outcome), whereas the (normalized) TBNN correctly reproduces distributional features observed in DNS, capturing complex interplay between strain and pressure fields.

This demonstrates that capturing alignment statistics is more meaningful for maintaining physical realism in turbulence modeling than minimizing pointwise tensor error.

4. Physical Insights and Practical Findings

Several key observations emerge:

• The original RFDM typically yields positive or negative definite pressure–Hessian, generating strong (and nonphysical) alignment biases with the strain–rate eigenvectors.
• The modified TBNN, utilizing normalized bases, yields pressure–Hessian tensors whose eigenvector PDFs closely track those of the true DNS field. This holds for both isotropic turbulence at $Re=433, 315$ and, to an extent, even for channel flows at higher $Re$.
• In striking contrast to common neural closure approaches, nearly all coefficients $C^i$ of the tensor basis are discovered to be almost constant (low variance) across the dataset. Thus, the full map $P_{tf}$ can be represented as a fixed linear combination of the normalized bases—a marked simplification with significant implications for turbulence closure methods.

A plausible implication is that, given careful construction of feature normalization, closure models for turbulence (e.g., in Lagrangian PDF approaches) can leverage such constant-coefficient tensor basis representations to capture non-local pressure effects using only local gradient information.

5. Limitations and Future Prospects

Notable limitations include:

• The original (strictly invariant) TBNN—without basis normalization—fails to reproduce correct alignment statistics, indicating that invariance constraints alone are insufficient when local intermittency is essential.
• While eigenvector statistics transfer well from isotropic to more complex flows, elementwise mean error rises significantly in anisotropic wall-bounded flows, showing that wall or boundary effects, and possibly explicit non-local terms, may be critical for pointwise accuracy.
• The restriction to local velocity gradient input omits the influence of truly nonlocal pressure terms, a known limitation in turbulence closures; future models may benefit by incorporating multiscale or nonlocal statistics.
• The current training is limited to isotropic turbulence and moderate $Re$. Extensions to higher $Re$, transitional, or multiphysics flows (e.g., compressible or buoyancy-driven) would require additional development.

Directions for future research include exploring invariance-preserving normalization strategies, integrating constant-coefficient basis models into Lagrangian PDF solvers, and incorporating flow history or explicitly nonlocal features.

6. Broader Impact on Turbulence Closure and Modeling

This work demonstrates that deep tensor basis networks, designed with physical symmetries and carefully tuned normalization, can serve as accurate surrogates for nonlocal pressure-velocity coupling in turbulent flows. The resulting constant-coefficient tensor basis models provide a practical and physically justifiable approach for pressure–Hessian closure, with the potential to improve high-fidelity turbulence simulations and subgrid modeling in both Lagrangian and Eulerian frameworks.

These findings indicate that neural velocity flow networks, when built as TBNNs with physically motivated features and normalization, offer both a mechanistically transparent and computationally efficient tool for turbulence closure—substantiated by strong statistical agreement (alignment statistics) with ground truth and by robust numerical performance across varied flow scenarios (Parashar et al., 2019).