SearcharxivSearch

arXiv subjects

Will Trojak

Publications and source records attributed to Will Trojak.

At least 19 recordsLinked to original sources

Precipitation Downscaling Using Foundation Model-Conditioned Diffusion

High-resolution precipitation fields are essential for hydrological impact assessment, yet global climate model outputs are too coarse and biased for direct use. AI-based statistical downscaling with diffusion models offers a promising approach, but the mechanism by which large-scale atmospheric predictors condition generation remains largely unexplored. We investigate three conditioning strategies for a denoising diffusion probabilistic model applied to daily precipitation downscaling: channel concatenation of upsampled coarse predictors, cross-attention conditioning with a learned convolutional encoder, and cross-attention conditioning with the frozen encoder of the pretrained Prithvi WxC weather foundation model. All strategies are evaluated against an unconditioned baseline under identical conditions using probabilistic, distributional, spectral, and extreme-event metrics for the Colorado River Basin. Concatenation conditioning achieves the lowest point-wise CRPS and MSE, but tends to produce over-smoothed fields that suppress high-intensity events. In contrast, cross-attention conditioning provides substantially better distributional realism and modest improvements in spectral fidelity. Improvements are greatest for extremes: the Prithvi-WxC conditioned model retains over half of >100mm/day events, although estimates are uncertain due to limited samples. When trained on the full dataset, the learned convolutional model performs similarly to the foundation model-conditioned approach while requiring lower computational resources. However, the Prithvi-WxC-conditioned model achieves comparable performance with only five years of training data. These results indicate that cross-attention conditioning offers advantages over simple concatenation for probabilistic precipitation downscaling, and that pre-trained foundation model representations may offer benefits in data-limited settings.

cs.CV

Direct numerical simulations of the supersonic Taylor--Green vortex via the Boltzmann equation

We explore the dynamics of the three-dimensional compressible Taylor--Green vortex from the perspective of kinetic theory by directly solving the six-dimensional Boltzmann equation. This work studies the connections between molecular-scale information encoded in the high-dimensional distribution function (e.g., molecular entropy measures) and macroscopic turbulent flow characteristics. We present high-order direct numerical simulations at Mach numbers of $0.5$ to $1.25$ and Reynolds numbers of $400$ to $1600$ performed using up to $137{\times}10^{9}$ degrees of freedom. The results indicate that the Kullback--Leibler divergence of the distribution function $f$ and its local equilibrium state $M[f]$ (i.e., the relative entropy functional $\langle f \log (f/M[f]) \rangle$) is strongly related to the macroscopic viscous dissipation rate, with the relative entropy value matching the sum of the solenoidal and dilatational dissipation rates very closely across a range of Mach and Reynolds numbers. Furthermore, we present the behavior of subgrid-scale quantities under spatial averaging/filtering operations performed directly on the particle distribution function, which shows notable similarities between the subgrid-scale dissipation and subgrid-scale relative entropy. These observations imply that information encoded in the distribution function, particularly certain measures of its deviation from equilibrium, may be useful for closure-modeling for compressible turbulence.

physics.flu-dyn

Machine Learning with Physics Knowledge for Prediction: A Survey

This survey examines the broad suite of methods and models for combining machine learning with physics knowledge for prediction and forecast, with a focus on partial differential equations. These methods have attracted significant interest due to their potential impact on advancing scientific research and industrial practices by improving predictive models with small- or large-scale datasets and expressive predictive models with useful inductive biases. The survey has two parts. The first considers incorporating physics knowledge on an architectural level through objective functions, structured predictive models, and data augmentation. The second considers data as physics knowledge, which motivates looking at multi-task, meta, and contextual learning as an alternative approach to incorporating physics knowledge in a data-driven fashion. Finally, we also provide an industrial perspective on the application of these methods and a survey of the open-source ecosystem for physics-informed machine learning.

cs.LG

Prithvi WxC: Foundation Model for Weather and Climate

Triggered by the realization that AI emulators can rival the performance of traditional numerical weather prediction models running on HPC systems, there is now an increasing number of large AI models that address use cases such as forecasting, downscaling, or nowcasting. While the parallel developments in the AI literature focus on foundation models -- models that can be effectively tuned to address multiple, different use cases -- the developments on the weather and climate side largely focus on single-use cases with particular emphasis on mid-range forecasting. We close this gap by introducing Prithvi WxC, a 2.3 billion parameter foundation model developed using 160 variables from the Modern-Era Retrospective Analysis for Research and Applications, Version 2 (MERRA-2). Prithvi WxC employs an encoder-decoder-based architecture, incorporating concepts from various recent transformer models to effectively capture both regional and global dependencies in the input data. The model has been designed to accommodate large token counts to model weather phenomena in different topologies at fine resolutions. Furthermore, it is trained with a mixed objective that combines the paradigms of masked reconstruction with forecasting. We test the model on a set of challenging downstream tasks namely: Autoregressive rollout forecasting, Downscaling, Gravity wave flux parameterization, and Extreme events estimation. The pretrained model with 2.3 billion parameters, along with the associated fine-tuning workflows, has been publicly released as an open-source contribution via Hugging Face.

cs.LG

PyFR v2.0.3: Towards Industrial Adoption of Scale-Resolving Simulations

PyFR is an open-source cross-platform computational fluid dynamics framework based on the high-order Flux Reconstruction approach, specifically designed for undertaking high-accuracy scale-resolving simulations in the vicinity of complex engineering geometries. Since the initial release of PyFR v0.1.0 in 2013, a range of new capabilities have been added to the framework, with a view to enabling industrial adoption of the capability. This paper provides details of those enhancements as released in PyFR v2.0.3, explains efforts to grow an engaged developer and user community, and provides latest performance and scaling results on up to 1024 AMD Instinct MI250X accelerators of Frontier at ORNL (each with two GCDs), and up to 2048 NVIDIA GH200 GPUs on Alps at CSCS.

physics.comp-ph

On the anti-aliasing properties of entropy filtering for discontinuous spectral element approximations of under-resolved turbulent flows

For large Reynolds number flows, it is typically necessary to perform simulations that are under-resolved with respect to the underlying flow physics. For nodal discontinuous spectral element approximations of these under-resolved flows, the collocation projection of the nonlinear flux can introduce aliasing errors which can result in numerical instabilities. In Dzanic and Witherden (J. Comput. Phys., 468, 2022), an entropy-based adaptive filtering approach was introduced as a robust, parameter-free shock-capturing method for discontinuous spectral element methods. This work explores the ability of entropy filtering for mitigating aliasing-driven instabilities in the simulation of under-resolved turbulent flows through high-order implicit large eddy simulations of a NACA0021 airfoil in deep stall at a Reynolds number of 270,000. It was observed that entropy filtering can adequately mitigate aliasing-driven instabilities without degrading the accuracy of the underlying high-order scheme on par with standard anti-aliasing methods such as over-integration, albeit with marginally worse performance at higher approximation orders.

math.NA

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving, parameter-free numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive, parameter-free switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier--Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.

physics.flu-dyn

Probing optimisation in physics-informed neural networks

A novel comparison is presented of the effect of optimiser choice on the accuracy of physics-informed neural networks (PINNs). To give insight into why some optimisers are better, a new approach is proposed that tracks the training trajectory curvature and can be evaluated on the fly at a low computational cost. The linear advection equation is studied for several advective velocities, and we show that the optimiser choice substantially impacts PINNs model performance and accuracy. Furthermore, using the curvature measure, we found a negative correlation between the convergence error and the curvature in the optimiser local reference frame. It is concluded that, in this case, larger local curvature values result in better solutions. Consequently, optimisation of PINNs is made more difficult as minima are in highly curved regions.

cs.LG

Bounds Preserving Temporal Integration Methods for Hyperbolic Conservation Laws

In this work, we present a modification of explicit Runge-Kutta temporal integration schemes that guarantees the preservation of any locally-defined quasiconvex set of bounds for the solution. These schemes operate on the basis of a bijective mapping between an admissible set of solutions and the real domain to strictly enforce bounds. Within this framework, we show that it is possible to recover a wide range of methods independently of the spatial discretization, including positivity preserving, discrete maximum principle satisfying, entropy dissipative, and invariant domain preserving schemes. Furthermore, these schemes are proven to recover the order of accuracy of the underlying Runge-Kutta method upon which they are built. The additional computational cost is the evaluation of two nonlinear mappings which generally have closed-form solutions. We show the utility of this approach in numerical experiments using a pseudospectral spatial discretization without any explicit shock capturing schemes for nonlinear hyperbolic problems with discontinuities.

math.NA

An extended range of stable flux reconstruction schemes on quadrilaterals for various polynomial bases

An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals -- which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.

math.NA

Utilizing Time-Reversibility for Shock Capturing in Nonlinear Hyperbolic Conservation Laws

In this work, we introduce a novel approach to formulating an artificial viscosity for shock capturing in nonlinear hyperbolic systems by utilizing the property that the solutions of hyperbolic conservation laws are not reversible in time in the vicinity of shocks. The proposed approach does not require any additional governing equations or a priori knowledge of the hyperbolic system in question, is independent of the mesh and approximation order, and requires the use of only one tunable parameter. The primary novelty is that the resulting artificial viscosity is unique for each component of the conservation law which is advantageous for systems in which some components exhibit discontinuities while others do not. The efficacy of the method is shown in numerical experiments of multi-dimensional hyperbolic conservation laws such as nonlinear transport, Euler equations, and ideal magnetohydrodynamics using a high-order discontinuous spectral element method on unstructured grids.

math.NA

An extended range of energy stable flux reconstruction methods on triangles

We present an extended range of stable flux reconstruction (FR) methods on triangles through the development and application of the summation-by-parts framework in two-dimensions. This extended range of stable schemes is then shown to contain the single parameter schemes of \citet{Castonguay2011} on triangles, and our definition enables wider stability bounds to be developed for those single parameter families. Stable upwinded spectral difference (SD) schemes on triangular elements have previously been found using Fourier analysis. We used our extended range of FR schemes to investigate the linear stability of SD methods on triangles, and it was found that a only first order SD scheme could be recovered within this set of FR methods.

math.NA

Nonlinear p-multigrid preconditioner for implicit time integration of compressible Navier--Stokes equations

Within the framework of $ p $-adaptive flux reconstruction, we aim to construct efficient polynomial multigrid ($p$MG) preconditioners for implicit time integration of the Navier--Stokes equations using Jacobian-free Newton--Krylov (JFNK) methods. We hypothesise that in pseudo transient continuation (PTC), as the residual drops, the frequency of error modes that dictates the convergence rate gets higher and higher. We apply nonlinear $p$MG solvers to stiff steady problems at low Mach number ($\mathrm{Ma}=10^{-3}$) to verify our hypothesis. It is demonstrated that once the residual drops by a few orders of magnitude, improved smoothing on intermediate $ p $-sublevels will not only maintain the stability of $ p $MG at large time steps but also improve the convergence rate. For the unsteady Navier--Stokes equations, we elaborate how to construct nonlinear preconditioners using pseudo transient continuation for the matrix-free generalized minimal residual (GMRES) method used in explicit first stage, singly diagonally implicit Runge--Kutta (ESDIRK) methods, and linearly implicit Rosenbrock--Wanner (ROW) methods. Given that at each time step the initial guess in the nonlinear solver is not distant from the converged solution, we recommend a two-level $p\{p_0\text{-}p_0/2\} $ or even $ p\{p_0\text{-}(p_0-1)\} $ $p$-hierarchy for optimal efficiency with a matrix-based smoother on the coarser level based on our hypothesis. It is demonstrated that insufficient smoothing on intermediate $p$-sublevels will deteriorate the performance of $p$MG preconditioner greatly. (See full abstract in the paper.)

math.NA

On the Use of RBF Interpolation for Flux Reconstruction

Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss--Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.

math.NA

Artificial Compressibility Approaches in Flux Reconstruction for Incompressible Viscous Flow Simulations

Several competing artificial compressibility methods for the incompressible flow equations are examined using the high-order flux reconstruction method. The established artificial compressibility method (ACM) of \citet{Chorin1967} is compared to the alternative entropically damped (EDAC) method of \citet{Clausen2013}, as well as an ACM formulation with hyperbolised diffusion. While the former requires the solution to be converged to a divergence free state at each physical time step through pseudo iterations, the latter can be applied explicitly. We examine the sensitivity of both methods to the parameterisation for a series of test cases over a range of Reynolds numbers. As the compressibility is reduced, EDAC is found to give linear improvements in divergence whereas ACM yields diminishing returns. For the Taylor--Green vortex, EDAC is found to perform well; however on the more challenging circular cylinder at $Re=3900$, EDAC gives rise to early transition of the free shear-layer and over-production of the turbulence kinetic energy. This is attributed to the spatial pressure fluctuations of the method. Similar behaviour is observed for an aerofoil at $Re=60,000$ with an attached transitional boundary layer. It is concluded that hyperbolic diffusion of ACM can be beneficial but at the cost of case setup time, and EDAC can be an efficient method for incompressible flow. However, care must be taken as pressure fluctuations can have a significant impact on physics and the remedy causes the governing equation to become overly stiff.

physics.flu-dyn

Hyperbolic Diffusion in Flux Reconstruction: Optimisation through Kernel Fusion within Tensor-Product Elements

Novel methods are presented in this initial study for the fusion of GPU kernels in the artificial compressibility method (ACM), using tensor product elements with constant Jacobians and flux reconstruction. This is made possible through the hyperbolisation of the diffusion terms, which eliminates the expensive algorithmic steps needed to form the viscous stresses. Two fusion approaches are presented, which offer differing levels of parallelism. This is found to be necessary for the change in workload as the order of accuracy of the elements is increased. Several further optimisations of these approaches are demonstrated, including a generation time memory manager which maximises resource usage. The fused kernels are able to achieve 3-4 times speedup, which compares favourably with a theoretical maximum speedup of 4. In three dimensional test cases, the generated fused kernels are found to reduce total runtime by ${\sim}25\%$, and, when compared to the standard ACM formulation, simulations demonstrate that a speedup of $2.3$ times can be achieved.

cs.MS

A Riemann Difference Scheme for Shock Capturing in Discontinuous Finite Element Methods

We present a novel structure-preserving numerical scheme for discontinuous finite element approximations of nonlinear hyperbolic systems. The method can be understood as a generalization of the Lax-Friedrichs flux to a high-order staggered grid and does not depend on any tunable parameters. Under a presented set of conditions, we show that the method is conservative and invariant domain preserving. Numerical experiments on the Euler equations show the ability of the scheme to resolve discontinuities without introducing excessive spurious oscillations or dissipation.

math.NA

Accuracy, Stability, and Performance Comparison between the Spectral Difference and Flux Reconstruction Schemes

We report the development of a discontinuous spectral element flow solver that includes the implementation of both spectral difference and flux reconstruction formulations. With this high order framework, we have constructed a foundation upon which to provide a fair and accurate assessment of these two schemes in terms of accuracy, stability, and performance with special attention to the true spectral difference scheme and the modified spectral difference scheme recovered via the flux reconstruction formulation. Building on previous analysis of the spectral difference and flux reconstruction schemes, we provide a novel nonlinear stability analysis of the spectral difference scheme. Through various numerical experiments, we demonstrate the additional stability afforded by the true, baseline spectral difference scheme without explicit filtering or de-aliasing due to its inherent feature of staggered flux points. This arrangement leads to favorable suppression of aliasing errors and improves stability needed for under-resolved simulations of turbulent flows.

physics.comp-ph