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Mohammad Farazmand

Publications and source records attributed to Mohammad Farazmand.

At least 19 recordsLinked to original sources
arXiv2026

Slow Manifold Reduction for Inertial Particles with Quadratic Drag

We consider the dynamics of inertial particles in unsteady fluid flows. At low Reynolds numbers, where the drag force is linear in the relative velocity, it is well-known that the dynamics admit an attracting, invariant, slow manifold which emerges as the perturbation of a normally hyperbolic critical manifold. However, at high Reynolds numbers, where the drag force is quadratic in the relative velocity, the critical manifold is no longer normally hyperbolic, and therefore its persistence has remained an open problem. Here, we resolve this issue by a particular application of the blowup method, which transforms the equations of motion to a generalized weighted cylindrical coordinate system, thereby desingularizing the dynamics on the critical manifold. We subsequently prove that the critical manifold persists under sufficiently small perturbations and derive the reduced equations of motion on the perturbed slow manifold to arbitrary accuracy. Our reduced equation differs from its linear-drag counterpart in its asymptotic expansion as well as its convergence rate. Using two examples, we demonstrate the validity of our slow manifold reduction. We also showcase an application of the reduced equations to the problem of source inversion in a turbulent dispersion model.

math.DS↗
arXiv2026

A review of shape-morphing solutions and evolutional neural networks for spatiotemporal dynamics

Shape-morphing solutions (SMS) refer to a class of approximate solutions of partial differential equations (PDEs) with the distinguishing feature that they depend nonlinearly on a set of time-dependent parameters. They generalize Galerkin truncations by allowing the basis (or trial functions) to evolve in time in order to adapt to the solution of the PDE. As such, SMS are particularly suitable for reduced-order modeling as well as high fidelity simulation of multiscale systems which exhibit localized time-dependent features, such as vortices, dispersive wave packets, and shocks. Furthermore, being mesh-free, SMS is scalable for solving PDEs in higher spatial dimensions. As a special case, SMS allows the approximation of the PDE's solution by a neural network whose weights and biases depend on time. Such neural networks are known as evolutional neural networks or neural Galerkin schemes. The evolution of SMS parameters is dictated by the SMS equation, a set of ordinary differential equations derived from the Dirac-Frenkel variational principle. Over the past five years, contributions to the theory and computation of SMS have been growing rapidly. Here, we survey these developments, showcase some applications of SMS, and highlight important open problems for future research. At the same time, this review is structured to serve as a tutorial for applied mathematicians, physicist, and engineers who wish to enter this field.

math.NA↗
arXiv2026

Rapid estimation of global sea surface temperatures from sparse streaming in situ observations

Reconstructing high-resolution sea surface temperatures (SST) from staggered SST measurements is essential for weather forecasting and climate projections. However, when SST measurements are sparse, the resulting inferred SST fields are rather inaccurate. Here, we demonstrate the ability of Sparse Discrete Empirical Interpolation Method (S-DEIM) to reconstruct the high-resolution SST field from sparse in situ observations, without using a model. The S-DEIM estimate consists of two terms, one computed from instantaneous in situ observations using empirical interpolation, and the other learned from the historical time series of observations using recurrent neural networks (RNNs). We train the RNNs using the National Oceanic and Atmospheric Administration's weekly high-resolution SST dataset spanning the years 1989-2021 which constitutes the training data. Subsequently, we examine the performance of S-DEIM on the test data, comprising January 2022 to January 2023. For this test data, S-DEIM infers the high-resolution SST from 100 in situ observations, constituting only 0.2% of the high-resolution spatial grid. We show that the resulting S-DEIM reconstructions are about 40% more accurate than earlier empirical interpolation methods, such as DEIM and Q-DEIM. Furthermore, 91% of S-DEIM estimates fall within $\pm 1^\circ$C of the true SST. We also demonstrate that S-DEIM is robust with respect to sensor placement: even when the sensors are distributed randomly, S-DEIM reconstruction error deteriorates only by 1-2%. S-DEIM is also computationally efficient. Training the RNN, which is performed only once offline, takes approximately one minute. Once trained, the S-DEIM reconstructions are computed in less than a second. As such, S-DEIM can be used for rapid SST reconstruction from sparse streaming observational data in real time.

physics.ao-ph↗
arXiv2025

Discrete Empirical Interpolation Method with Upper and Lower Bound Constraints

Discrete Empirical Interpolation Method (DEIM) is a simple and effective method for reconstructing a function from its incomplete pointwise observations. However, applying DEIM to functions with physically constrained ranges can produce reconstructions with values outside the prescribed physical bounds. Such physically constrained quantities occur routinely in applications, e.g., mass density whose range is nonnegative. The DEIM reconstructions which violate these physical constraints are not usable in downstream tasks such as forecasting and control. To address this issue, we develop Constrained DEIM (C-DEIM) whose reconstructions are guaranteed to respect the physical bounds of the quantity of interest. C-DEIM enforces the bounds as soft constraints, in the form of a carefully designed penalty term, added to the underlying least squares problem. We prove that the C-DEIM reconstructions satisfy the physical constraints asymptotically, i.e., as the penalty parameter increases towards infinity. We also derive a quantitative upper bound for the observation residual of C-DEIM. Based on these theoretical results, we devise an efficient algorithm for practical implementation of C-DEIM. The efficacy of the method and the accompanying algorithm are demonstrated on several examples, including a heat transfer problem from fluid dynamics and a cellular automaton model of wildfire spread.

math.NA↗
arXiv2025

Bridging the Gap Between Deterministic and Probabilistic Approaches to State Estimation

We consider the problem of state estimation from limited discrete and noisy measurements. In particular, we focus on modal state estimation, which approximates the unknown state of the system within a prescribed basis. We estimate the coefficients of the modal expansion using available observational data. This is usually accomplished through two distinct frameworks. One is deterministic and estimates the expansion coefficients by solving a least-squares (LS) problem. The second is probabilistic and uses a Bayesian approach to derive a distribution for the coefficients, resulting in the maximum-a-posteriori (MAP) estimate. Here, we seek to quantify and compare the accuracy of these two approaches. To this end, we derive a computable expression for the difference in Bayes risk between the deterministic LS and the Bayesian MAP estimates. We prove that this difference is always nonnegative, indicating that the MAP estimate is always more reliable than the LS estimate. We further show that this difference comprises two nonnegative components representing measurement noise and prior uncertainty, and identify regimes where one component dominates the other in magnitude. We also derive a novel prior distribution from the sample covariance matrix of the training data, and examine the greedy Bayesian and column-pivoted QR (CPQR) sensor placement algorithms with this prior as an input. Using numerical examples, we show that the greedy Bayesian algorithm returns nearly optimal sensor locations. We show that, under certain conditions, the greedy Bayesian sensor locations are identical or nearly identical to those of CPQR when applied to a regularized modal basis.

math.NA↗
arXiv2024

Sequential data assimilation for PDEs using shape-morphing solutions

Shape-morphing solutions (also known as evolutional deep neural networks, reduced-order nonlinear solutions, and neural Galerkin schemes) are a new class of methods for approximating the solution of time-dependent partial differential equations (PDEs). Here, we introduce a sequential data assimilation method for incorporating observational data in a shape-morphing solution (SMS). Our method takes the form of a predictor-corrector scheme, where the observations are used to correct the SMS parameters using Newton-like iterations. Between observation points, the SMS equations (a set of ordinary differential equations) are used to evolve the solution forward in time. We prove that, under certain conditions, the data assimilated SMS (DA-SMS) converges uniformly towards the true state of the system. We demonstrate the efficacy of DA-SMS on three examples: the nonlinear Schrodinger equation, the Kuramoto-Sivashinsky equation, and a two-dimensional advection-diffusion equation. Our numerical results suggest that DA-SMS converges with relatively sparse observations and a single iteration of the Newton-like method.

math.NA↗
arXiv2024

Extreme firebrand transport by atmospheric waves in wildfires

In wildfires, burning pieces of ember-firebrands-are carried downstream by wind. At the time of landing, these firebrands can start secondary fires far away from the main burning unit. This phenomenon is called spotting and the secondary fires are referred to as spot fires. Here, we first present numerical evidence that atmospheric traveling waves can increase the spotting distance by at least an order of magnitude compared to unidirectional wind conditions. We then present theoretical results explaining this numerical observation. In particular, we show that the firebrand's motion can synchronize with the traveling wave, leading to a surf-like motion for some firebrand particles. This delays the firebrand's landing, making extreme spotting distances possible. This physical phenomena helps explain the discrepancy between previous theoretical estimates of maximum spotting distance and much larger spotting distances observed empirically. We derive new analytical expressions for the landing time and landing distance of the firebrands.

physics.flu-dyn↗
arXiv2024

State Estimation Using Sparse DEIM and Recurrent Neural Networks

Sparse Discrete Empirical Interpolation Method (S-DEIM) was recently proposed for state estimation in dynamical systems when only a sparse subset of the state variables can be observed. The S-DEIM estimate involves a kernel vector whose optimal value is inferred through a data assimilation algorithm. This data assimilation step suffers from two drawbacks: (i) It requires the knowledge of the governing equations of the dynamical system, and (ii) It is not generally guaranteed to converge to the optimal kernel vector. To address these issues, here we introduce an equation-free S-DEIM framework that estimates the optimal kernel vector from sparse observational time series using recurrent neural networks (RNNs). We show that the recurrent architecture is necessary since the kernel vector cannot be estimated from instantaneous observations. But RNNs, which incorporate the past history of the observations in the learning process, lead to nearly optimal estimations. We demonstrate the efficacy of our method on three numerical examples with increasing degree of spatiotemporal complexity: a conceptual model of atmospheric flow known as the Lorenz-96 system, the Kuramoto-Sivashinsky equation, and the Rayleigh-Benard convection. In each case, the resulting S-DEIM estimates are satisfactory even when a relatively simple RNN architecture, namely the reservoir computing network, is used. More specifically, our RNN-based S-DEIM state estimations reduce the relative error between 42% and 58% when compared to Q-DEIM which ignores the kernel vector by setting it equal to zero.

math.DS↗
arXiv2024

Sparse Discrete Empirical Interpolation Method: State Estimation from Few Sensors

Discrete empirical interpolation method (DEIM) estimates a function from its incomplete pointwise measurements. Unfortunately, DEIM suffers large interpolation errors when few measurements are available. Here, we introduce Sparse DEIM (S-DEIM) for accurately estimating a function even when very few measurements are available. To this end, S-DEIM leverages a kernel vector which has been neglected in previous DEIM-based methods. We derive theoretical error estimates for S-DEIM, showing its relatively small error when an optimal kernel vector is used. When the function is generated by a continuous-time dynamical system, we propose a data assimilation algorithm which approximates the optimal kernel vector using observational time series. We prove that, under certain conditions, data assimilated S-DEIM converges exponentially fast towards the true state. We demonstrate the efficacy of our method on two numerical examples.

math.NA↗
arXiv2023

Enforcing conserved quantities in Galerkin truncation and finite volume discretization

Finite-dimensional truncations are routinely used to approximate partial differential equations (PDEs), either to obtain numerical solutions or to derive reduced-order models. The resulting discretized equations are known to violate certain physical properties of the system. In particular, first integrals of the PDE may not remain invariant after discretization. Here, we use the method of reduced-order nonlinear solutions (RONS) to ensure that the conserved quantities of the PDE survive its finite-dimensional truncation. In particular, we develop two methods: Galerkin RONS and finite volume RONS. Galerkin RONS ensures the conservation of first integrals in Galerkin-type truncations, whether used for direct numerical simulations or reduced-order modeling. Similarly, finite volume RONS conserves any number of first integrals of the system, including its total energy, after finite volume discretization. Both methods are applicable to general time-dependent PDEs and can be easily incorporated in existing Galerkin-type or finite volume code. We demonstrate the efficacy of our methods on two examples: direct numerical simulations of the shallow water equation and a reduced-order model of the nonlinear Schrodinger equation. As a byproduct, we also generalize RONS to phenomena described by a system of PDEs.

math.NA↗
arXiv2023

Modeling correlated uncertainties in stochastic compartmental models

We consider compartmental models of communicable disease with uncertain contact rates. Stochastic fluctuations are often added to the contact rate to account for uncertainties. White noise, which is the typical choice for the fluctuations, leads to significant underestimation of the disease severity. Here, starting from reasonable assumptions on the social behavior of individuals, we model the contacts as a Markov process which takes into account the temporal correlations present in human social activities. Consequently, we show that the mean-reverting Ornstein-Uhlenbeck (OU) process is the correct model for the stochastic contact rate. We demonstrate the implication of our model on two examples: a Susceptibles-Infected-Susceptibles (SIS) model and a Susceptibles-Exposed-Infected-Removed (SEIR) model of the COVID-19 pandemic. In particular, we observe that both compartmental models with white noise uncertainties undergo transitions that lead to the systematic underestimation of the spread of the disease. In contrast, modeling the contact rate with the OU process significantly hinders such unrealistic noise-induced transitions. For the SIS model, we derive its stationary probability density analytically, for both white and correlated noise. This allows us to give a complete description of the model's asymptotic behavior as a function of its bifurcation parameters, i.e., the basic reproduction number, noise intensity, and correlation time. For the SEIR model, where the probability density is not available in closed form, we study the transitions using Monte Carlo simulations. Our study underscores the necessity of temporal correlations in stochastic compartmental models and the need for more empirical studies that would systematically quantify such correlations.

q-bio.PE↗
arXiv2023

Fisher information and shape-morphing modes for solving the Fokker-Planck equation in higher dimensions

The Fokker-Planck equation describes the evolution of the probability density associated with a stochastic differential equation. As the dimension of the system grows, solving this partial differential equation (PDE) using conventional numerical methods becomes computationally prohibitive. Here, we introduce a fast, scalable, and interpretable method for solving the Fokker-Planck equation which is applicable in higher dimensions. This method approximates the solution as a linear combination of shape-morphing Gaussians with time-dependent means and covariances. These parameters evolve according to the method of reduced-order nonlinear solutions (RONS) which ensures that the approximate solution stays close to the true solution of the PDE for all times. As such, the proposed method approximates the transient dynamics as well as the equilibrium density, when the latter exists. Our approximate solutions can be viewed as an evolution on a finite-dimensional statistical manifold embedded in the space of probability densities. We show that the metric tensor in RONS coincides with the Fisher information matrix on this manifold. We also discuss the interpretation of our method as a shallow neural network with Gaussian activation functions and time-varying parameters. In contrast to existing deep learning methods, our method is interpretable, requires no training, and automatically ensures that the approximate solution satisfies all properties of a probability density.

math.DS↗
arXiv2023

Fast and scalable computation of shape-morphing nonlinear solutions with application to evolutional neural networks

We develop fast and scalable methods for computing reduced-order nonlinear solutions (RONS). RONS was recently proposed as a framework for reduced-order modeling of time-dependent partial differential equations (PDEs), where the modes depend nonlinearly on a set of time-varying parameters. RONS uses a set of ordinary differential equations (ODEs) for the parameters to optimally evolve the shape of the modes to adapt to the PDE's solution. This method has already proven extremely effective in tackling challenging problems such as advection-dominated flows and high-dimensional PDEs. However, as the number of parameters grow, integrating the RONS equation and even its formation become computationally prohibitive. Here, we develop three separate methods to address these computational bottlenecks: symbolic RONS, collocation RONS and regularized RONS. We demonstrate the efficacy of these methods on two examples: Fokker-Planck equation in high dimensions and the Kuramoto-Sivashinsky equation. In both cases, we observe that the proposed methods lead to several orders of magnitude in speedup and accuracy. Our proposed methods extend the applicability of RONS beyond reduced-order modeling by making it possible to use RONS for accurate numerical solution of linear and nonlinear PDEs. Finally, as a special case of RONS, we discuss its application to problems where the PDE's solution is approximated by a neural network, with the time-dependent parameters being the weights and biases of the network. The RONS equations dictate the optimal evolution of the network's parameters without requiring any training.

math.DS↗
arXiv2023

Tensor-based flow reconstruction from optimally located sensor measurements

Reconstructing high-resolution flow fields from sparse measurements is a major challenge in fluid dynamics. Existing methods often vectorize the flow by stacking different spatial directions on top of each other, hence confounding the information encoded in different dimensions. Here, we introduce a tensor-based sensor placement and flow reconstruction method which retains and exploits the inherent multidimensionality of the flow. We derive estimates for the flow reconstruction error, storage requirements and computational cost of our method. We show, with examples, that our tensor-based method is significantly more accurate than similar vectorized methods. Furthermore, the variance of the error is smaller when using our tensor-based method. While the computational cost of our method is comparable to similar vectorized methods, it reduces the storage cost by several orders of magnitude. The reduced storage cost becomes even more pronounced as the dimension of the flow increases. We demonstrate the efficacy of our method on three examples: a chaotic Kolmogorov flow, in-situ and satellite measurements of the global sea surface temperature, and 3D unsteady simulated flow around a marine research vessel.

physics.flu-dyn↗
arXiv2022

Stochastic compartmental models of COVID-19 pandemic must have temporally correlated uncertainties

Compartmental models are an important quantitative tool in epidemiology, enabling us to forecast the course of a communicable disease. However, the model parameters, such as the infectivity rate of the disease, are riddled with uncertainties, which has motivated the development and use of stochastic compartmental models. Here, we first show that a common stochastic model, which treats the uncertainties as white noise, is fundamentally flawed since it erroneously implies that greater parameter uncertainties will lead to the eradication of the disease. Then, we present a principled modeling of the uncertainties based on reasonable assumptions on the contacts of each individual. Using the central limit theorem and Doob's theorem on Gaussian Markov processes, we prove that the correlated Ornstein-Uhlenbeck process is the appropriate tool for modeling uncertainties in the infectivity rate. We demonstrate our results using a compartmental model of the COVID-19 pandemic and the available US data from the Johns Hopkins University COVID-19 database. In particular, we show that the white noise stochastic model systematically underestimates the severity of the Omicron variant of COVID-19, whereas the Ornstein-Uhlenbeck model correctly forecasts the course of this variant. Moreover, using an SIS model of sexually transmitted disease, we derive an exact closed-form solution for the asymptotic distribution of infected individuals. This analytic result shows that the white noise model underestimates the severity of the pandemic because of unrealistic noise-induced transitions. Our results strongly support the need for temporal correlations in modeling of uncertainties in compartmental models of infectious disease.

q-bio.PE↗
arXiv2022

Quantifying rare events in spotting: How far do wildfires spread?

Spotting refers to the transport of burning pieces of firebrand by wind which, at the time of landing, may ignite new fires beyond the direct ignition zone of the main fire. Spot fires that occur far from the original burn unit are rare but have consequential ramifications since their prediction and control remains challenging. To facilitate their prediction, we examine three methods for quantifying the landing distribution of firebrands: crude Monte Carlo simulations, importance sampling, and large deviation theory (LDT). In particular, we propose an LDT method that accurately and parsimoniously quantifies the low probability events at the tail of the landing distribution. In contrast, Monte Carlo and importance sampling methods are most efficient in quantifying the high probability landing distances near the mode of the distribution. However, they become computationally intractable for quantifying the tail of the distribution due to the large sample size required. We also show that the most probable landing distance grows linearly with the mean characteristic velocity of the wind field. Furthermore, defining the relative landed mass as the proportion of mass landed at a given distance from the main fire, we derive an explicit formula which allows computing this quantity as a function of the landing distribution at a negligible computational cost. We numerically demonstrate our findings on two prescribed wind fields.

physics.data-an↗
arXiv2022

Shape-morphing reduced-order models for nonlinear Schrodinger equations

We consider reduced-order modeling of nonlinear dispersive waves described by a class of nonlinear Schrodinger (NLS) equations. We compare two nonlinear reduced-order modeling methods: (i) The reduced Lagrangian approach which relies on the variational formulation of NLS and (ii) The recently developed method of reduced-order nonlinear solutions (RONS). First, we prove the surprising result that, although the two methods are seemingly quite different, they can be obtained from the real and imaginary parts of a single complex-valued master equation. Furthermore, for the NLS equation in a stationary frame, we show that the reduced Lagrangian method fails to predict the correct group velocity of the waves whereas RONS predicts the correct group velocity. Finally, for the modified NLS equation, where the reduced Lagrangian approach is inapplicable, the RONS reduced-order model accurately approximates the true solutions.

math.DS↗
arXiv2022

Model-assisted deep learning of rare extreme events from partial observations

To predict rare extreme events using deep neural networks, one encounters the so-called small data problem because even long-term observations often contain few extreme events. Here, we investigate a model-assisted framework where the training data is obtained from numerical simulations, as opposed to observations, with adequate samples from extreme events. However, to ensure the trained networks are applicable in practice, the training is not performed on the full simulation data; instead we only use a small subset of observable quantities which can be measured in practice. We investigate the feasibility of this model-assisted framework on three different dynamical systems (Rossler attractor, FitzHugh-Nagumo model, and a turbulent fluid flow) and three different deep neural network architectures (feedforward, long short-term memory, and reservoir computing). In each case, we study the prediction accuracy, robustness to noise, reproducibility under repeated training, and sensitivity to the type of input data. In particular, we find long short-term memory networks to be most robust to noise and to yield relatively accurate predictions, while requiring minimal fine-tuning of the hyperparameters.

cs.LG↗