Averaging or filtering the Navier-Stokes equations leaves a term, the Reynolds stress in the Reynolds-averaged (RANS) equations or the subgrid stress in large-eddy simulation (LES), that cannot be computed from the resolved field alone. Decades of modelling have produced closures of great practical value, from the one-equation and two-equation eddy-viscosity models described by Wilcox (2006) to the subgrid model of Smagorinsky (1963) and its dynamic successors. Their weaknesses are equally well known: separated flows, strong streamline curvature, secondary motions and transition are predicted poorly by models calibrated on simple shear flows.

The growth of direct numerical simulation databases prompted the question of whether closures could be learned instead of derived. The reviews by Duraisamy et al. (2019) and Brunton et al. (2020) chart the first wave of that effort. A recurring finding was that unconstrained regression performs badly, because a network free to violate invariance or conservation will do so as soon as it leaves its training data. Hence the interest in methods that carry physics inside the learning problem.

Physics-informed neural networks (PINNs), introduced in their current form by Raissi et al. (2019), are the best known of these methods, and the phrase has become a banner for a broader movement (Karniadakis et al., 2021). This review examines what the PINN idea has contributed to turbulence closure, where it has run into trouble, and what kind of evidence would allow claims in this area to be compared.

This is a narrative review of the published literature, selective by design and concentrated on incompressible flow. A difficulty of terminology has to be settled first. In the strict sense a PINN is a network that represents the solution fields as functions of space and time and is trained by minimising the residuals of the governing equations, evaluated by automatic differentiation at collocation points, together with boundary and data terms. The idea of using a network as a trial solution of a differential equation is older than the name. In the turbulence literature, however, "physics-informed machine learning" was also adopted for something different: regression models for Reynolds-stress discrepancies whose inputs and outputs respect physical constraints (Wang et al., 2017). Both usages are covered here, and they are kept apart, because their strengths and failure modes differ.

The work reviewed falls into three groups according to where the physics enters and where the closure lives. Figure 1 lays out the grouping, and the subsections below follow it.

Enthusiasm for these methods has been tempered by a set of difficulties that appear again and again. Table 1 lists them with the sources that analyse each; the paragraphs below comment on those that bear most directly on closure.

The first three are properties of PINN training in general. The loss combines terms with different units and very different gradient magnitudes, and Wang et al. (2021) showed that the imbalance can prevent the boundary conditions from being learned at all. Analysis through the neural tangent kernel explains why components of the solution converge at widely separated rates (Wang et al., 2022), which is one face of the preference of networks for low frequencies documented by Rahaman et al. (2019). Krishnapriyan et al. (2021) demonstrated that the standard formulation can fail on simple convection and reaction problems, and that the cause is the optimisation landscape created by the soft constraint. Turbulent flows, with thin near-wall layers and a broad range of scales, exercise every one of these weaknesses, which is why published turbulent applications remain at modest Reynolds numbers and in simple geometries (Cai et al., 2021).

The fourth and fifth are specific to closure and more fundamental. Wu et al. (2019) showed that the RANS equations with an explicitly prescribed Reynolds stress can be ill-conditioned: even stresses taken directly from direct numerical simulation, with errors of a size no model could avoid, may yield mean velocities that are badly wrong. Treating part of the stress implicitly through an eddy viscosity restores conditioning. The lesson generalises. A model judged by how well it reproduces stresses on held-out snapshots, an a priori test, tells little about the flow it will produce once coupled to a solver. For LES the same gap appears as numerical instability when a network that correlates well with the true subgrid stress is inserted into a simulation (Beck et al., 2019).

Two further matters are raised less often than they should be. One is cost. A PINN that needs hours of training on a graphics processor to solve one steady RANS problem has to be compared with a finite-volume solver that reaches the same answer in minutes, and few papers report that comparison. The other is uncertainty. A learned closure applied outside its training range gives no warning that it is extrapolating, whereas the limits of the classical models, although real, are at least well mapped.

Progress in learned closures is hard to assess because studies seldom share test cases, baselines or metrics. Each paper tends to introduce its own flow, its own training split and its own measure of error, and to compare against a single conventional model run with unstated settings. Other areas of machine learning advanced quickly once common benchmarks existed, and turbulence modelling has a long tradition of validation cases on which to build.

Several directions look productive. Hard enforcement of constraints, whether by building invariance into the architecture as the tensor-basis network does or by training through a solver that satisfies conservation exactly, has a better record than soft penalties for closure work. Hybrid schemes that use a PINN for what it does well, assimilating sparse measurements into a field consistent with the equations, and a conventional solver for prediction, play to the strengths of each. Interpretable outputs, algebraic expressions or corrections to named terms of an existing model, ease adoption because they can be inspected and implemented in production codes. The strict PINN as a general-purpose turbulent flow solver with an embedded closure remains, on present evidence, a research topic and not yet a competitor to established methods.

Physics-informed learning has changed how the closure problem is approached. It has produced invariant stress models, inversion-based corrections and solver-in-the-loop training, each with documented successes on separated and secondary flows that trouble classical closures. The strict PINN formulation has contributed most where data are sparse and the task is reconstruction, and least where a transferable predictive closure is wanted. Its training difficulties are now well characterised. What the field lacks is shared evidence. A tiered, open benchmark with a posteriori metrics, honest baselines and reported cost would do more to separate durable advances from fragile ones than another architecture.

To be declared by the author(s) before publication.

To be declared by the author(s) before publication.