arXiv subjects

Saad Waheed

Publications and source records attributed to Saad Waheed.

6 recordsLinked to original sources

arXiv2026

Learning Structural Eigenmodes with Modal Operator Network (ModalONet)

Neural operators such as the deep operator network (DeepONet) and the Laplace neural operator (LNO) are effective surrogates, but they are almost exclusively trained to reproduce the forward response of a system rather than its intrinsic structure, such as the modal properties of a structural system. We introduce ModalONet, which puts operator learning to a different use, such as recovering the modal basis, namely: mode shapes, natural frequencies, and damping ratios of a dynamical system directly from its response field, with no eigensolver and no labeled modes. Our key observation is that the DeepONet branch-trunk factorization is itself a learnable form of modal superposition: the trunk supplies continuous, mesh-free mode shapes, while an LNO branch supplies the modal coordinates in pole-residue form, so that each learned pole yields a natural frequency and a damping ratio. Training uses the response field alone, under a composite loss of reconstruction, orthonormality, and temporal projection consistency. The degenerate (equal-frequency) modes are resolved by a separable trunk and a shared frequency parameter, with post-hoc log-envelope regression improves the damping ratio estimates. Across four structural systems, namely: simply supported and cantilever Euler-Bernoulli beams and rectangular and (degenerate) square Kirchhoff plates, the ModalONet recovers the analytical modal basis with modal assurance criterion (MAC) values of at least 0.998 for every mode shape, natural frequency errors within 5%, and damping ratio errors within 7%, demonstrating the potential of neural operators as accurate and interpretable tools for modal identification.

eess.SY↗

arXiv2026

Koopman Spectral Analysis of Lithium-Ion Battery Dynamics: State of Charge as a Marginally Stable Observable

Accurate state-of-charge (SOC) estimation remains a fundamental challenge in lithium-ion battery management systems because battery dynamics are highly nonlinear, operating-condition dependent, and sensitive to parameter variations caused by aging and temperature. Conventional model-based estimators, such as equivalent circuit model (ECM) and Kalman-filter-based approaches, rely heavily on repeated parameter identification and accurate electrochemical modeling, whereas purely data-driven methods often sacrifice physical interpretability. This work proposes a Koopman-theoretic, data-driven framework for SOC estimation using Dynamic Mode Decomposition with control (DMDc) combined with Hankel time-delay embedding. Instead of explicitly identifying ECM parameters, the proposed approach reconstructs a lifted dynamical state space directly from measured terminal voltage and current obtained through Hybrid Pulse Power Characterization (HPPC) testing. Spectral decomposition of the identified DMDc operator reveals intrinsic battery dynamics in terms of Koopman modes and eigenvalues. The SOC dynamics naturally emerge as the slowest marginally stable mode whose eigenvalue lies closest to the unit circle, consistent with the integrator-type behavior of charge conservation. The corresponding modal coordinate is subsequently utilized as an SOC-sensitive observable.

eess.SY↗

arXiv2026

A Graph Theoretic Approach in Combination With Dynamic Mode Decomposition With Control (DMDc) to Analyze Battery Degradation

Accurate monitoring of lithium-ion battery (LIB) degradation is essential, yet it remains challenging due to the complex, nonlinear, and time-varying nature of electrochemical aging processes. Conventional equivalent circuit models (ECMs) provide simplified representations of battery behavior using fixed electrical components, but they cannot capture evolving internal degradation mechanisms and structural changes over time. In this study, a data-driven framework is developed by integrating dynamic mode decomposition with control (DMDc) with graph-theoretic analysis to characterize battery degradation from operational data alone. The mode matrix ($\mathbf{\phi}$) obtained from DMDc is transformed into a weighted adjacency matrix, enabling the representation of battery dynamics as an evolving network of interacting states. Graph-based measures, including connectivity and a modularity (proxy), are then used to quantify structural changes in the system across degradation stages. The results show a clear transition from a highly connected and coherent network in the healthy state to a progressively weaker and more fragmented structure as degradation advances, accompanied by increasing heterogeneity. This work demonstrates that graph-theoretic representations can effectively capture the evolving dynamics of battery degradation and provide interpretable insights into system-level aging behavior.

eess.SY↗

arXiv2025

The Limiting Spectral Distribution of Various Matrix Ensembles Under the Anticommutator Operation

Inspired by the quantization of classical quantities and Rankin Selberg convolution, we study the anticommutator operation $\{\cdot, \cdot\}$, where $\{A,B\} = AB + BA$, applied to real symmetric random matrix ensembles including Gaussian orthogonal ensemble (GOE), the palindromic Toeplitz ensemble (PTE), the $k$-checkerboard ensemble, and the block $k$-circulant ensemble ($k$-BCE). Using combinatorial and topological techniques related to non-crossing and free matching properties of GOE and PTE, we obtain closed-form formulae for the moments of the limiting spectral distributions of $\{$GOE, GOE$\}$, $\{$PTE, PTE$\}$, $\{$GOE, PTE$\}$ and establish the corresponding limiting spectral distributions with generating functions and convolution. On the other hand, $\{$GOE, $k$-checkerboard$\}$ and $\{$$k$-checkerboard, $j$-checkerboard$\}$ exhibit entirely different spectral behavior than the other anticommutator ensembles: while the spectrum of $\{$GOE, $k$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$ and 1 blip regime of size $\Theta(N^{3/2})$, the spectrum of $\{$$k$-checkerboard, $j$-checkerboard$\}$ consists of 1 bulk regime of size $\Theta(N)$, 2 intermediary blip regimes of size $\Theta(N^{3/2})$, and 1 largest blip regime of size $\Theta(N^2)$. In both cases, with the appropriate weight function, we are able to isolate the largest regime for other regime(s) and analyze its moments and convergence results via combinatorics. We end with numerical computation of lower even moments of $\{$GOE, $k$-BCE$\}$ and $\{$$k$-BCE, $k$-BCE$\}$ based on genus expansion and discussion on the challenge with analyzing the intermediary blip regimes of $\{$$k$-checkerboard, $j$-checkerboard$\}$.

math.PR↗

arXiv2024

Leslie Population Models in Predator-prey and Competitive populations: theory and applications by machine learning

We introduce a new predator-prey model by replacing the growth and predation constant by a square matrix, and the population density as a population vector. The classical Lotka-Volterra model describes a population that either modulates or converges. Stability analysis of such models have been extensively studied by the works of Merdan (https://doi.org/10.1016/j.chaos.2007.06.062). The new model adds complexity by introducing an age group structure where the population of each age group evolves as prescribed by the Leslie matrix. The added complexity changes the behavior of the model such that the population either displays roughly an exponential growth or decay. We first provide an exact equation that describes a time evolution and use analytic techniques to obtain an approximate growth factor. We also discuss the variants of the Leslie model, i.e., the complex value predator-prey model and the competitive model. We then prove the Last Species Standing theorem that determines the dominant population in the large time limit. The recursive structure of the model denies the application of simple regression. We discuss a machine learning scheme that allows an admissible fit for the population evolution of Paramecium Aurelia and Paramecium Caudatum. Another potential avenue to simplify the computation is to use the machinery of quantum operators. We demonstrate the potential of this approach by computing the Hamiltonian of a simple Leslie system.

q-bio.PE↗

arXiv2024

A closed formula for linear recurrences with constant coefficients

Given a linear recurrence of the form $c_n=a_1c_{n-1}+\cdots+a_j c_{n-j}$, it is well-known that $c_n=\sum_{r}p_r(n)r^n$, where the sum is taken over the set of characteristic roots and each $p_r(n)$ is some polynomial. We give a closed formula for the coefficients of each polynomial $p_r(n)$ for any linear recurrence of this form.

math.CO↗