arXiv subjects

Inanc Senocak

Publications and source records attributed to Inanc Senocak.

At least 19 recordsLinked to original sources

arXiv2025

Unsupervised simulation of incompressible flows with physics- and equality- constrained artificial neural networks

Physics-informed neural networks (PINNs) have shown promise for solving partial differential equations, yet their success in simulating incompressible flows at high Reynolds numbers remains limited. Existing approaches rely on auxiliary labeled data, supervised pretraining, or reference solutions, and no purely unsupervised method comparable to conventional finite-difference or finite-volume solvers has been demonstrated. We attribute this gap to the absence of a mechanism for enforcing the divergence-free constraint and boundary conditions to strict tolerances. To address this, we adopt the physics- and equality-constrained artificial neural network (PECANN) framework with a conditionally adaptive augmented Lagrangian method (CA-ALM), and introduce a pressure-Poisson-based objective. The residual of the pressure Poisson equation is minimized subject to the momentum and continuity equations and boundary conditions on the primitive variables as equality constraints, with CA-ALM enforcing all constraints tightly. For advection-dominated, high-Reynolds-number flows, we further propose an adaptive vanishing entropy viscosity that stabilizes early training without influencing the converged solution. A baseline that instead uses the momentum residual as the objective proves ineffective under the same machinery, underscoring the critical role of the pressure-Poisson objective. The method is assessed on lid-driven cavity flow up to $Re=7{,}500$, three-dimensional unsteady Beltrami flow, and steady and unsteady flow past a circular cylinder with general inflow-outflow boundary conditions, including an ablation study identifying admissible outlet conditions -- all without labeled data or supervised pretraining. Notably, it captures the spontaneous onset of periodic vortex shedding in unsteady cylinder flow without external perturbations, starting from a randomly initialized network.

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arXiv2025

Long-lived versus nocturnal stable atmospheric boundary layers: DNS characterisation, similarity theory, and regime classification

The stable atmospheric boundary layer (SABL) is broadly classified into two subtypes: the nocturnal SABL, driven primarily by surface cooling, and the long-lived SABL, in which ambient stratification coexists with surface cooling and persists well beyond a single diurnal cycle. Despite its prevalence in polar regions and over open oceans, the long-lived SABL has received comparatively little attention, and a systematic direct numerical simulation (DNS) study distinguishing it from the nocturnal SABL has not previously been reported. Here we present DNS of long-lived SABLs at $Re_D=900$, contrasting weak and strong ambient stratification cases against a nocturnal SABL. A striking feature of the long-lived SABLs is the emergence of a multi-layered thermal structure -- including an intermediate layer of reduced static stability capped by a buoyancy inversion -- driven not by enhanced turbulent mixing, but by a stratification-induced reorganisation of the buoyancy field. Budget analyses identify a turbulent potential energy-centred transport mechanism absent in nocturnal SABLs, lending support to total-energy-based closure approaches. Assessment of an extended Monin--Obukhov similarity theory incorporating a composite length scale $L_*$ shows strong collapse of dimensionless gradients of velocity $\phi_m$ and potential temperature $\phi_h$, and a new similarity function for $\phi_h$ is proposed. Combining DNS with linear stability analysis, we construct a regime map delineating linearly stable, weakly stable, and very stable regimes within a two-parameter dimensionless space, demonstrating the inherently multi-parameter nature of long-lived SABLs and the limitations of single-parameter subgrid-scale parameterisations.

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arXiv2025

Conditionally adaptive augmented Lagrangian method for physics-informed learning of forward and inverse problems

We present several key advances to the Physics and Equality Constrained Artificial Neural Networks (PECANN) framework, substantially improving its capacity to solve challenging partial differential equations (PDEs). Our enhancements broaden the framework's applicability and improve efficiency. First, we generalize the Augmented Lagrangian Method (ALM) to support multiple, independent penalty parameters for enforcing heterogeneous constraints. Second, we introduce a constraint aggregation technique to address inefficiencies associated with point-wise enforcement. Third, we incorporate a single Fourier feature mapping to capture highly oscillatory solutions with multi-scale features, where alternative methods often require multiple mappings or costlier architectures. Fourth, a novel time-windowing strategy enables seamless long-time evolution without relying on discrete time models. Fifth, and critically, we propose a conditionally adaptive penalty update (CAPU) strategy for ALM that accelerates the growth of Lagrange multipliers for constraints with larger violations, while enabling coordinated updates of multiple penalty parameters. CAPU accelerates the growth of Lagrange multipliers for selectively challenging constraints, enhancing constraint enforcement during training. We demonstrate the effectiveness of PECANN-CAPU across diverse problems, including the transonic rarefaction problem, reversible scalar advection by a vortex, high-wavenumber Helmholtz and Poisson's equations, and inverse heat source identification. The framework achieves competitive accuracy across all cases when compared with established methods and recent approaches based on Kolmogorov-Arnold networks. Collectively, these advances improve the robustness, computational efficiency, and applicability of PECANN to demanding problems in scientific computing.

cs.LG↗

arXiv2025

Dynamical stability and flow regimes in a stably stratified valley-shaped cavity heated from below

We investigate the three-dimensional stability of a stably stratified fluid in a valley-shaped cavity heated from below using linear stability analysis and direct numerical simulations. We first describe the pure-conduction flow state and derive a dimensionless criterion that provides a lower bound for the onset of instability, valid for any slope angle. We then examine the sequence of flow regimes for a slope angle of $\alpha = 30^{\circ}$ and Prandtl number $Pr = 7$, including two-dimensional steady states, the emergence of a Hopf bifurcation, and the formation of steady and oscillatory three-dimensional structures preceding the transition to fully unsteady, chaotic flow. Although the nonlinear governing equations depend on two dimensionless parameters, we find that the flow dynamics across a wide parameter range collapse to depend on a single parameter--the composite stratification parameter $\Pi_c$. However, as the system becomes more unstable, sensitivity to the second parameter, $\Pi_h$, increases. We construct a regime map of all observed flow states as a function of $\Pi_c$ and $\Pi_h$, and confirm the onset of chaos using Lyapunov exponents. Across all regimes, asymmetric circulation remains the dominant flow structure, persisting even in time-averaged fields of chaotic states. Finally, we characterize heat transfer in the cavity using the Nusselt number, which scales as $Nu \sim \Pi_c^{0.43}$ or equivalently $Nu \sim Ra^{0.275}$. This result further establishes $\Pi_c$ as a key dimensionless parameter governing the flow dynamics preceding the chaotic regime.

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arXiv2024

Non-overlapping, Schwarz-type Domain Decomposition Method for Physics and Equality Constrained Artificial Neural Networks

We present a non-overlapping, Schwarz-type domain decomposition method with a generalized interface condition, designed for physics-informed machine learning of partial differential equations (PDEs) in both forward and inverse contexts. Our approach employs physics and equality-constrained artificial neural networks (PECANN) within each subdomain. Unlike the original PECANN method, which relies solely on initial and boundary conditions to constrain PDEs, our method uses both boundary conditions and the governing PDE to constrain a unique interface loss function for each subdomain. This modification improves the learning of subdomain-specific interface parameters while reducing communication overhead by delaying information exchange between neighboring subdomains. To address the constrained optimization in each subdomain, we apply an augmented Lagrangian method with a conditionally adaptive update strategy, transforming the problem into an unconstrained dual optimization. A distinct advantage of our domain decomposition method is its ability to learn solutions to both Poisson's and Helmholtz equations, even in cases with high-wavenumber and complex-valued solutions. Through numerical experiments with up to 64 subdomains, we demonstrate that our method consistently generalizes well as the number of subdomains increases.

cs.LG↗

arXiv2023

Self-organization in a stably stratified, valley-shaped enclosure heated from below

We observe the spontaneous onset of three-dimensional motion from a quiescent, purely conductive state of a stably stratified fluid in a V-shaped enclosure heated from below, which ultimately self-organizes into a two-dimensional steady state without any external forcing to the initial configuration. We identify a dominant three-dimensional instability through modal stability analysis. Direct numerical simulations confirm this instability but also reveal that, after an initial period of spontaneous three-dimensional growth, the flow gradually self-organizes into a steady two-dimensional state without external intervention. This self-organization manifests consistently for any arbitrary infinitesimal three-dimensional disturbance to the initial quiescent configuration. We demonstrate that the mechanism driving this self-organization is the increasing dominance of viscous dissipation over buoyant production of disturbance kinetic energy at later stages of flow evolution from the initial quiescent state. Our investigation reveals a flow scenario in which the most natural transition pathway to the final state involves passing through an intermediate state with a higher dimension than the final state itself. Specifically, our final flow state is less complex than the three-dimensional most unstable eigenvector predicted by linear stability analysis. We demonstrate that the entire flow evolution remains non-turbulent throughout and closely aligns with results from linear stability analysis, distinguishing the present flow dynamics from transient chaos, which also features complex transient states that eventually converge to a less complex final state.

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arXiv2023

A Generalized Schwarz-type Non-overlapping Domain Decomposition Method using Physics-constrained Neural Networks

We present a meshless Schwarz-type non-overlapping domain decomposition method based on artificial neural networks for solving forward and inverse problems involving partial differential equations (PDEs). To ensure the consistency of solutions across neighboring subdomains, we adopt a generalized Robin-type interface condition, assigning unique Robin parameters to each subdomain. These subdomain-specific Robin parameters are learned to minimize the mismatch on the Robin interface condition, facilitating efficient information exchange during training. Our method is applicable to both the Laplace's and Helmholtz equations. It represents local solutions by an independent neural network model which is trained to minimize the loss on the governing PDE while strictly enforcing boundary and interface conditions through an augmented Lagrangian formalism. A key strength of our method lies in its ability to learn a Robin parameter for each subdomain, thereby enhancing information exchange with its neighboring subdomains. We observe that the learned Robin parameters adapt to the local behavior of the solution, domain partitioning and subdomain location relative to the overall domain. Extensive experiments on forward and inverse problems, including one-way and two-way decompositions with crosspoints, demonstrate the versatility and performance of our proposed approach.

cs.LG↗

arXiv2023

An adaptive augmented Lagrangian method for training physics and equality constrained artificial neural networks

Physics and equality constrained artificial neural networks (PECANN) are grounded in methods of constrained optimization to properly constrain the solution of partial differential equations (PDEs) with their boundary and initial conditions and any high-fidelity data that may be available. To this end, adoption of the augmented Lagrangian method within the PECANN framework is paramount for learning the solution of PDEs without manually balancing the individual loss terms in the objective function used for determining the parameters of the neural network. Generally speaking, ALM combines the merits of the penalty and Lagrange multiplier methods while avoiding the ill conditioning and convergence issues associated singly with these methods . In the present work, we apply our PECANN framework to solve forward and inverse problems that have an expanded and diverse set of constraints. We show that ALM with its conventional formulation to update its penalty parameter and Lagrange multipliers stalls for such challenging problems. To address this issue, we propose an adaptive ALM in which each constraint is assigned a unique penalty parameter that evolve adaptively according to a rule inspired by the adaptive subgradient method. Additionally, we revise our PECANN formulation for improved computational efficiency and savings which allows for mini-batch training. We demonstrate the efficacy of our proposed approach by solving several forward and PDE-constrained inverse problems with noisy data, including simulation of incompressible fluid flows with a primitive-variables formulation of the Navier-Stokes equations up to a Reynolds number of 1000.

cs.LG↗

arXiv2023

Instabilities of longitudinal vortex rolls in katabatic Prandtl slope flows

Stationary counter-rotating longitudinal vortex pairs emerge from one-dimensional Prandtl slope flows under katabatic as well as anabatic conditions due to a linear instability when the imposed surface heat flux magnitude is sufficiently strong relative to the stable ambient stratification. For anabatic flows, these vortices have already been identified to exhibit an unique topology that bears a striking resemblance to speaker-wires since they stay coherent as a single unit without the presence of another vortex pair. Under katabatic conditions and at a constant Prandtl number, we find that the longitudinal vortices emerging at a range of different slope angles possess the similar topology as their anabatic counterparts. We determine the existence of both fundamental and subharmonic secondary instabilities depending on the slope angle for the most likely transverse base flow wavelength. Our results indicate that the most dominant instability shifts from a fundamental to subharmonic mode with increasing slope angle. At shallow slopes, this dynamic contrast with the speaker-wire vortices in anabatic slope flows at the same angle which for which the subharmonic instability is clearly dominant. These modes are responsible for the bending and movement of single or multiple speaker-wire vortices, which may merge or reconnect to lead to dynamically more unstable states, eventually leading to transition towards turbulence. We demonstrate that at sufficiently steep slopes, the dynamics of these vortex pairs are dominated by long-wave reconnections or two-dimensional mergers between adjacent pairs.

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arXiv2023

An unusual bifurcation scenario in a stably stratified, valley-shaped enclosure heated from below

We delineate the structure of steady laminar flows within a stably stratified, valley-shaped triangular cavity heated from below through linear stability analysis and Navier-Stokes simulations. We derive an exact solution to the quiescent conduction state, and characterize the flow via the stratification perturbation parameter, $\Pi_s$, which is a measure of the strength of the surface heat flux relative to the background stable stratification. Beyond a threshold value of $\Pi_s$, two unstable eigenmodes appear, one marked by a dominant central circulation, and the other one exhibiting dual circulations of equal strength. Through Navier-Stokes simulations, we confirm that the central-circulation eigenmode generates a pair of asymmetric steady states, whereas the dual-circulation eigenmode leads to distinct upslope and downslope symmetric steady states. Linear stability analysis and Navier-Stokes simulations jointly confirm the instability of the two symmetric steady states, both of which transition to the asymmetric steady state under a perturbation. Thus, for a given set of dimensionless parameters, the Navier-Stokes equations admit at least five possible steady-state solutions. Two of these solutions, namely the quiescent, pure conduction state and the counter-intuitive symmetric downslope state, have previously been overlooked in heated, stably stratified, valley-shaped enclosures. These five flow solutions reveal an intriguing bifurcation structure, including both a perfect pitchfork bifurcation and a nested bifurcation that gives rise to two distinct states. The inner bifurcation, while resembling a pitchfork in some respects, does not break any symmetry of the valley due to the lack of any possible horizontal axis of symmetry. The categorization of this inner bifurcation remains an unresolved matter, as it does not conform to any established descriptions of canonical bifurcations.

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arXiv2022

Critical Investigation of Failure Modes in Physics-informed Neural Networks

Several recent works in scientific machine learning have revived interest in the application of neural networks to partial differential equations (PDEs). A popular approach is to aggregate the residual form of the governing PDE and its boundary conditions as soft penalties into a composite objective/loss function for training neural networks, which is commonly referred to as physics-informed neural networks (PINNs). In the present study, we visualize the loss landscapes and distributions of learned parameters and explain the ways this particular formulation of the objective function may hinder or even prevent convergence when dealing with challenging target solutions. We construct a purely data-driven loss function composed of both the boundary loss and the domain loss. Using this data-driven loss function and, separately, a physics-informed loss function, we then train two neural network models with the same architecture. We show that incomparable scales between boundary and domain loss terms are the culprit behind the poor performance. Additionally, we assess the performance of both approaches on two elliptic problems with increasingly complex target solutions. Based on our analysis of their loss landscapes and learned parameter distributions, we observe that a physics-informed neural network with a composite objective function formulation produces highly non-convex loss surfaces that are difficult to optimize and are more prone to the problem of vanishing gradients.

cs.LG↗

arXiv2022

Characterizing and Mitigating the Difficulty in Training Physics-informed Artificial Neural Networks under Pointwise Constraints

Neural networks can be used to learn the solution of partial differential equations (PDEs) on arbitrary domains without requiring a computational mesh. Common approaches integrate differential operators in training neural networks using a structured loss function. The most common training algorithm for neural networks is backpropagation which relies on the gradient of the loss function with respect to the parameters of the network. In this work, we characterize the difficulty of training neural networks on physics by investigating the impact of differential operators in corrupting the back propagated gradients. Particularly, we show that perturbations present in the output of a neural network model during early stages of training lead to higher levels of noise in a structured loss function that is composed of high-order differential operators. These perturbations consequently corrupt the back-propagated gradients and impede convergence. We mitigate this issue by introducing auxiliary flux parameters to obtain a system of first-order differential equations. We formulate a non-linear unconstrained optimization problem using the augmented Lagrangian method that properly constrains the boundary conditions and adaptively focus on regions of higher gradients that are difficult to learn. We apply our approach to learn the solution of various benchmark PDE problems and demonstrate orders of magnitude improvement over existing approaches.

cs.LG↗

arXiv2022

Speaker-wire vortices in stratified anabatic Prandtl slope flows and their secondary instabilities

Stationary longitudinal vortical rolls emerge in katabatic and anabatic Prandtl slope flows due to the dominance of the normal component of the buoyancy force over flow shear. Here, we further identify self pairing of these longitudinal rolls as a unique flow structure. The topology of the counter-rotating vortex pair bears a striking resemblance to speaker-wires and their interaction with each other is a precursor to further destabilization and breakdown of the flow field into smaller structures. On its own, a speaker-wire vortex retains its unique topology without any vortex reconnection or breakup. For a fixed slope angle $\alpha=3^{\circ}$ and at a constant Prandtl number, we analyse the saturated state of speaker-wire vortices and perform a bi-global linear stability analysis based on their stationary state. We establish the existence of both fundamental and subharmonic secondary instabilities depending on the circulation and transverse wavelength of the base state of speaker-wire vortices. The dominance of subharmonic modes relative to the fundamental mode helps explain the relative stability of a single vortex pair compared to the vortex dynamics in presence of two or an even number of pairs.These instability modes are essential for the bending and merging of multiple speaker-wire vortices, which break up and lead to more dynamically unstable states, eventually paving the way for transition towards turbulence. This process is demonstrated via direct numerical simulations with which we are able to track the nonlinear temporal evolution of these instabilities.

physics.flu-dyn↗

arXiv2021

Physics and Equality Constrained Artificial Neural Networks: Application to Forward and Inverse Problems with Multi-fidelity Data Fusion

Physics-informed neural networks (PINNs) have been proposed to learn the solution of partial differential equations (PDE). In PINNs, the residual form of the PDE of interest and its boundary conditions are lumped into a composite objective function as soft penalties. Here, we show that this specific way of formulating the objective function is the source of severe limitations in the PINN approach when applied to different kinds of PDEs. To address these limitations, we propose a versatile framework based on a constrained optimization problem formulation, where we use the augmented Lagrangian method (ALM) to constrain the solution of a PDE with its boundary conditions and any high-fidelity data that may be available. Our approach is adept at forward and inverse problems with multi-fidelity data fusion. We demonstrate the efficacy and versatility of our physics- and equality-constrained deep-learning framework by applying it to several forward and inverse problems involving multi-dimensional PDEs. Our framework achieves orders of magnitude improvements in accuracy levels in comparison with state-of-the-art physics-informed neural networks.

physics.comp-ph↗

arXiv2021

Impact of stratification mechanisms on turbulent characteristics of stable open-channel flows

Flow over a surface can be stratified by imposing a fixed mean vertical temperature (density) gradient profile throughout or via cooling at the surface. These distinct mechanisms can act simultaneously to establish a stable stratification in a flow. Here, we perform a series of direct numerical simulations of open-channel flows to study adaptation of a neutrally stratified turbulent flow under the combined or independent action of the aforementioned mechanisms. We force the fully developed flow with a constant mass flow rate. This flow forcing technique enables us to keep the bulk Reynolds number constant throughout our investigation and avoid complications arising from the acceleration of the bulk flow when a constant pressure gradient approach were to be adopted to force the flow instead. When both stratification mechanisms are active, the dimensionless stratification perturbation number emerges as an external flow control parameter, in addition to the Reynolds, Froude, and Prandtl numbers. We demonstrate that significant deviations from the Monin-Obukhov similarity formulation are possible when both types of stratification mechanisms are active within an otherwise weakly stable flow, even when the flux Richardson number is well below 0.2. An extended version of the similarity theory due to Zilitinkevich and Calanca shows promise in predicting the dimensionless shear for cases where both types of stratification mechanisms are active, but the extended theory is less accurate for gradients of scalar. The degree of deviation from neutral dimensionless shear as a function of the vertical coordinate emerges as a qualitative measure of the strength of stable stratification for all the cases investigated in this study.

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arXiv2019

Linear Stability of Katabatic Slope Flows with Ambient Wind Forcing

We investigate the stability of katabatic slope flows over an infinitely wide and uniformly cooled planar surface subject to an additional forcing due to a uniform downslope wind field aloft. We adopt an extension of Prandtl's original model for slope flows (Lykosov & Gutman 1972) to derive the base flow, which constitutes an interesting basic state in stability analysis because it cannot be reduced to a single universal form independent of external parameters. We apply a linear modal analysis to this basic state to demonstrate that for a fixed Prandtl number and slope angle, two independent dimensionless parameters are sufficient to describe the flow stability. One of these parameters is the stratification perturbation number that we have introduced in Xiao & Senocak (2019). The second parameter, which we will henceforth designate the wind forcing number, is hitherto uncharted and can be interpreted as the ratio of the kinetic energy of the ambient wind aloft to the damping due to viscosity and stabilizing effect of the background stratification. For a fixed Prandtl number, stationary transverse and travelling longitudinal modes of instabilities can emerge, depending on the value of the slope angle and the aforementioned dimensionless numbers. The influence of ambient wind forcing on the base flow's stability is complicated as the ambient wind can be both stabilizing as well as destabilizing for a certain range of the parameters. Our results constitute a strong counter-evidence against the current practice of relying solely on the gradient Richardson number to describe the dynamic stability of stratified atmospheric slope flows.

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arXiv2019

PittPack: An Open-Source Poisson's Equation Solver for Extreme-Scale Computing with Accelerators

We present a parallel implementation of a direct solver for the Poisson's equation on extreme-scale supercomputers with accelerators. We introduce a chunked-pencil decomposition as the domain-decomposition strategy to distribute work among processing elements to achieve superior scalability at large number of accelerators. Chunked-pencil decomposition enables overlapping nodal communication and data transfer between the central processing units (CPUs) and the graphics processing units (GPUs). Second, it improves data locality by keeping neighboring elements in adjacent memory locations. Third, it allows usage of shared-memory for certain segments of the algorithm when possible, and last but not least, it enables contiguous message transfer among the nodes. Two different communication patterns are designed. The fist pattern aims to fully overlap the communication with data transfer and designed for speedup of overall turnaround time, whereas the second method concentrates on low memory usage and is more network friendly for computations at extreme scale. To ensure software portability, we interleave OpenACC with MPI in the software. The numerical solution and its formal second order of accuracy is verified using method of manufactured solutions for various combinations of boundary conditions. Weak scaling analysis is performed using up to 1.1 trillion Cartesian mesh points using 16384 GPUs on a petascale leadership class supercomputer.

physics.comp-ph↗

arXiv2017

The role of bulk eddy-viscosity variation on the log-layer mismatch observed in wall-modeled large-eddy simulations

We investigate the role of the bulk eddy-viscosity variation on the well-known log-layer mismatch problem. An analysis of the mean momentum-balance shows that the modeled stress term close to the wall can dominate because of the bulk eddy-viscosity. Consequently, the momentum-balance equation lacks a degree-of-freedom and the mean velocity conforms to an incorrect profile to satisfy the momentum-balance. We show that zonal enforcement of the target mass flow-rate can be an effective strategy to introduce an additional degree of freedom to the mean momentum-balance, which led to a significant reduction in the log-layer mismatch. When the mass flow-rate is enforced zonally, the filtered velocity field attains its own constant velocity-scale above the Reynolds-averaged field, supporting the hypothesis that there exists an artificial boundary layer above the Reynolds-averaged region. We simulate turbulent channel flows at friction Reynolds numbers of 2000 and 5200 on coarse meshes that would put the first point away from the wall well into the logarithmic layer. Second-order turbulence statistics and one-dimensional velocity spectra agree well with the direct numerical simulation benchmark data when results are normalized by the velocity-scale extracted from the filtered velocity field. Additionally, the error in the skin-friction coefficient for friction Reynolds numbers of 2000 decreased from $14.1\%$ to $2.5\%$ when we enforced the mass flow-rate zonally.

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