arXiv subjects

Alessandro Della Pia

Publications and source records attributed to Alessandro Della Pia.

5 recordsLinked to original sources

arXiv2026

Autoencoders vs. Numerical Analysis--Informed Manifold Learning for Navier--Stokes Flows

Autoencoders (AEs) have become a dominant approach to nonlinear latent-space construction in data-driven reduced-order modelling (ROM), with their decoders lifting latent representations back to the ambient state space. Their prominence, however, has overshadowed an established alternative: manifold-learning methods grounded in classical numerical analysis. We revisit this alternative using Parsimonious Diffusion Maps (PDMs), benchmarking them against Proper Orthogonal Decomposition (POD)-based ROMs and several convolutional AE architectures for the two-dimensional incompressible flow past a rotating cylinder ---a bifurcating Navier-Stokes (NS) system organized by a codimension-2 Bogdanov-Takens point and its associated Hopf, saddle-node, and homoclinic bifurcations. Our approach uses PDMs to identify a parsimonious and interpretable set of intrinsic latent coordinates and to estimate their dimension directly from data. Gaussian process regression then learns the latent dynamics, while convex K-nearest-neighbor (K-NN) interpolation in PDMs space constructs the pre-image map, for which we establish pointwise consistency. The resulting nonlinear ROM substantially outperforms POD-based ROMs and achieves reconstruction and prediction accuracy comparable to ---and, in some bifurcating regimes, better than--- that of AE-based ROMs. At the same time, latent-variable learning with PDMs requires orders of magnitude less computational time than AE training.

math.NA↗

arXiv2026

Surrogate normal-forms for the numerical bifurcation and stability analysis of navier-stokes flows via machine learning

Inspired by the Equation-Free paradigm, we propose an ``embed-learn-lift'' framework for constructing minimal-dimensional surrogate ROMs for the numerical analysis of high-fidelity Navier-Stokes simulations, even in the presence of symmetries that standard machine-learning surrogates often fail to preserve. The framework consists of four main stages. First, manifold learning (here both POD and Diffusion Maps) is used to uncover the intrinsic geometry and dimensionality of the latent space underlying the high-dimensional spatio-temporal Navier-Stokes dynamics across the parameter space. Second, we construct ROMs (here, via Gaussian Process regression (GPR)) of minimal dimension -- by learning the evolution equations directly on the identified latent space. Third, we exploit the toolkit of numerical bifurcation analysis to construct bifurcation diagrams and perform systematic stability analysis directly in the latent coordinates. This enables, for example, the efficient continuation of branches of limit cycles emerging from Andronov-Hopf and Neimark-Sacker bifurcations, together with the computation of limit-cycles periods and stability properties via Floquet multipliers. Such analysis is effectively intractable for the full Navier-Stokes equations. Finally, by solving the pre-image problem in manifold learning, we reconstruct the bifurcating steady and time-periodic states in the original high-dimensional physical space, thus closing the ``lift'' step of the pipeline. We show that DMs-based ROMs allow for a computationally efficient and accurate numerical bifurcation and stability analysis, thus outperforming the widely used POD-ROMs by providing a geometrically consistent parametrization and correctly identifying the intrinsic dimension even in the presence of secondary instabilities, highlighting the need for nonlinear manifold learning methods in CFD.

physics.flu-dyn↗

arXiv2026

PDE-Free Mass-Constrained Learning of Complex Systems with Hidden States

We propose a three-tier machine learning framework based on the next-generation Equation-Free algorithm for learning the spatio-temporal dynamics of mass-constrained complex systems with hidden states, whose dynamics can in principle be described by PDEs, but lack explicit models. In the first step, we employ Diffusion Maps (DMs), a nonlinear manifold learning algorithm, to extract low-dimensional latent representations of the complex spatio-temporal evolution. In the second step, we learn manifold-informed reduced-order models (ROMs) with Sparse Identification of Nonlinear Dynamics (SINDy) and standard linear Multivariate Autoregressive models (MVARs) to approximate the solution operator on the latent space. In the final step, the latent dynamics are lifted back to the original high-dimensional space by solving an (ill-posed) pre-image problem via a convex interpolation based on the k-NN algorithm. In doing so, the proposed framework reconstructs the solution operator of the unknown mass-constrained PDE, without explicitly identifying the PDE itself. For comparison purposes, we also evaluated the performance of the scheme for constructing ROMs based on Proper Orthogonal Decomposition (POD) and prove that both POD and the k-NN lifting operators preserve the mass. We illustrate the approach using two benchmark problems: (a) the Hughes model of crowd dynamics, which minimizes walking time while avoiding obstacles and high-density regions, and (b) a CFD problem involving the spatio-temporal evolution of a passive tracer advected by a periodic Navier--Stokes velocity field. We show that ROMs informed by DMs yield parsimonious models that consistently outperform the best POD-informed ROMs, yielding stable and accurate approximations of the solution operator in the latent space and, via reconstruction, in the original high-dimensional space over long time horizons.

math.NA↗

arXiv2026

RANDSMAPs: Random-Feature/multi-Scale Neural Decoders with Mass Preservation

We introduce RANDSMAPs (Random-feature/multi-scale neural decoders with Mass Preservation), numerical analysis-informed, explainable neural decoders designed to explicitly respect conservation laws when solving the challenging ill-posed pre-image problem in manifold learning. We start by proving the equivalence of vanilla random Fourier feature neural networks to Radial Basis Function interpolation and the double Diffusion Maps (based on Geometric Harmonics) decoders in the deterministic limit. We then establish the theoretical foundations for RANDSMAP and introduce its multiscale variant to capture structures across multiple scales. We formulate and derive the closed-form solution of the corresponding constrained optimization problem and prove the mass preservation property. Numerically, we assess the performance of RANDSMAP on three benchmark problems/datasets with mass preservation obtained by the Lighthill-Whitham-Richards traffic flow PDE with shock waves, 2D rotated MRI brain images, and the Hughes crowd dynamics PDEs. We demonstrate that RANDSMAPs yield high reconstruction accuracy at low computational cost and maintain mass conservation at single-machine precision. In its vanilla formulation, the scheme remains applicable to the classical pre-image problem, i.e., when mass-preservation constraints are not imposed.

math.NA↗

arXiv2024

Learning the Latent dynamics of Fluid flows from High-Fidelity Numerical Simulations using Parsimonious Diffusion Maps

We use parsimonious diffusion maps (PDMs) to discover the latent dynamics of high-fidelity Navier-Stokes simulations with a focus on the 2D fluidic pinball problem. By varying the Reynolds number, different flow regimes emerge, ranging from steady symmetric flows to quasi-periodic asymmetric and turbulence. We show, that the proposed non-linear manifold learning scheme, identifies in a crisp manner the expected intrinsic dimension of the underlying emerging dynamics over the parameter space. In particular, PDMs, estimate that the emergent dynamics in the oscillatory regime can be captured by just two variables, while in the chaotic regime, the dominant modes are three as anticipated by the normal form theory. On the other hand, proper orthogonal decomposition (POD)/PCA, most commonly used for dimensionality reduction in fluid mechanics, does not provide such a crisp separation between the dominant modes. To validate the performance of PDMs, we also computed the reconstruction error, by constructing a decoder using Geometric Harmonics. We show that the proposed scheme outperforms the POD/PCA over the whole Reynolds number range. Thus, we believe that the proposed scheme will allow for the development of more accurate reduced order models for high-fidelity fluid dynamics simulators, thus relaxing the curse of dimensionality in numerical analysis tasks such as bifurcation analysis, optimization and control.

physics.flu-dyn↗