Ali Forootani

Ali Forootani, Pawan Goyal, Peter Benner

The discovery of governing equations from data has been an active field of research for decades. One widely used methodology for this purpose is sparse regression for nonlinear dynamics, known as SINDy. Despite several attempts, noisy and scarce data still pose a severe challenge to the success of the SINDy approach. In this work, we discuss a robust method to discover nonlinear governing equations from noisy and scarce data. To do this, we make use of neural networks to learn an implicit representation based on measurement data so that not only it produces the output in the vicinity of the measurements but also the time-evolution of output can be described by a dynamical system. Additionally, we learn such a dynamic system in the spirit of the SINDy framework. Leveraging the implicit representation using neural networks, we obtain the derivative information -- required for SINDy -- using an automatic differentiation tool. To enhance the robustness of our methodology, we further incorporate an integral condition on the output of the implicit networks. Furthermore, we extend our methodology to handle data collected from multiple initial conditions. We demonstrate the efficiency of the proposed methodology to discover governing equations under noisy and scarce data regimes by means of several examples and compare its performance with existing methods.

Ali Forootani, Mohammad Sadr, Danial Esmaeili Aliabadi et al.

Energy system models are increasingly employed to guide long-term planning in multi-sectoral environments where decisions span electricity, heat, transport, land use, and industry. While these models provide rigorous quantitative insights, their outputs are often highly technical, making them difficult to interpret for non-expert stakeholders such as policymakers, planners, and the public. This communication gap limits the accessibility and practical impact of scenario-based modeling, particularly as energy transitions grow more complex with rising shares of renewables, sectoral integration, and deep uncertainties. To address this challenge, we propose the Renewable Energy Large Language Model (RE-LLM), a hybrid framework that integrates Large Language Models (LLMs) directly into the energy system modeling workflow. RE-LLM combines three core elements: (i) optimization-based scenario exploration, (ii) machine learning surrogates that accelerate computationally intensive simulations, and (iii) LLM-powered natural language generation that translates complex results into clear, stakeholder-oriented explanations. This integrated design not only reduces computational burden but also enhances inter-pretability, enabling real-time reasoning about trade-offs, sensitivities, and policy implications. The framework is adaptable across different optimization platforms and energy system models, ensuring broad applicability beyond the case study presented. By merging speed, rigor, and interpretability, RE-LLM advances a new paradigm of human-centric energy modeling. It enables interactive, multilingual, and accessible engagement with future energy pathways, ultimately bridging the final gap between data-driven analysis and actionable decision-making for sustainable transitions.

Ali Forootani

Mathematical reasoning and optimization are fundamental to artificial intelligence and computational problem-solving. Recent advancements in Large Language Models (LLMs) have significantly improved AI-driven mathematical reasoning, theorem proving, and optimization techniques. This survey explores the evolution of mathematical problem-solving in AI, from early statistical learning approaches to modern deep learning and transformer-based methodologies. We review the capabilities of pretrained language models and LLMs in performing arithmetic operations, complex reasoning, theorem proving, and structured symbolic computation. A key focus is on how LLMs integrate with optimization and control frameworks, including mixed-integer programming, linear quadratic control, and multi-agent optimization strategies. We examine how LLMs assist in problem formulation, constraint generation, and heuristic search, bridging theoretical reasoning with practical applications. We also discuss enhancement techniques such as Chain-of-Thought reasoning, instruction tuning, and tool-augmented methods that improve LLM's problem-solving performance. Despite their progress, LLMs face challenges in numerical precision, logical consistency, and proof verification. Emerging trends such as hybrid neural-symbolic reasoning, structured prompt engineering, and multi-step self-correction aim to overcome these limitations. Future research should focus on interpretability, integration with domain-specific solvers, and improving the robustness of AI-driven decision-making. This survey offers a comprehensive review of the current landscape and future directions of mathematical reasoning and optimization with LLMs, with applications across engineering, finance, and scientific research.

Ali Forootani, Danial Esmaeili Aliabadi, Daniela Thraen

Wind power forecasting plays a critical role in modern energy systems, facilitating the integration of renewable energy sources into the power grid. Accurate prediction of wind energy output is essential for managing the inherent intermittency of wind power, optimizing energy dispatch, and ensuring grid stability. This paper proposes the use of Deep Neural Network (DNN)-based predictive models that leverage climate datasets, including wind speed, atmospheric pressure, temperature, and other meteorological variables, to improve the accuracy of wind power simulations. In particular, we focus on the Coupled Model Intercomparison Project (CMIP) datasets, which provide climate projections, as inputs for training the DNN models. These models aim to capture the complex nonlinear relationships between the CMIP-based climate data and actual wind power generation at wind farms located in Germany. Our study compares various DNN architectures, specifically Multilayer Perceptron (MLP), Long Short-Term Memory (LSTM) networks, and Transformer-enhanced LSTM models, to identify the best configuration among these architectures for climate-aware wind power simulation. The implementation of this framework involves the development of a Python package (CADNN) designed to support multiple tasks, including statistical analysis of the climate data, data visualization, preprocessing, DNN training, and performance evaluation. We demonstrate that the DNN models, when integrated with climate data, significantly enhance forecasting accuracy. This climate-aware approach offers a deeper understanding of the time-dependent climate patterns that influence wind power generation, providing more accurate predictions and making it adaptable to other geographical regions.

Ali Forootani, Raffaele Iervolino

Federated Learning (FL) has emerged as a powerful paradigm for decentralized machine learning, enabling collaborative model training across diverse clients without sharing raw data. However, traditional FL approaches often face limitations in scalability and efficiency due to their reliance on synchronous client updates, which can result in significant delays and increased communication overhead, particularly in heterogeneous and dynamic environments. To address these challenges in this paper, we propose an Asynchronous Federated Learning (AFL) algorithm, which allows clients to update the global model independently and asynchronously. Our key contributions include a comprehensive convergence analysis of AFL in the presence of client delays and model staleness. By leveraging martingale difference sequence theory and variance bounds, we ensure robust convergence despite asynchronous updates. Assuming strongly convex local objective functions, we establish bounds on gradient variance under random client sampling and derive a recursion formula quantifying the impact of client delays on convergence. Furthermore, we demonstrate the practical applicability of the AFL algorithm by training decentralized linear regression and Support Vector Machine (SVM) based classifiers and compare its results with synchronous FL algorithm to effectively handling non-IID data distributed among clients. The proposed AFL algorithm addresses key limitations of traditional FL methods, such as inefficiency due to global synchronization and susceptibility to client drift. It enhances scalability, robustness, and efficiency in real-world settings with heterogeneous client populations and dynamic network conditions. Our results underscore the potential of AFL to drive advancements indistributed learning systems, particularly for large-scale, privacy-preserving applications in resource-constrained environments.

Ali Forootani, Harshit Kapadia, Sridhar Chellappa et al.

The sparse identification of nonlinear dynamical systems (SINDy) is a data-driven technique employed for uncovering and representing the fundamental dynamics of intricate systems based on observational data. However, a primary obstacle in the discovery of models for nonlinear partial differential equations (PDEs) lies in addressing the challenges posed by the curse of dimensionality and large datasets. Consequently, the strategic selection of the most informative samples within a given dataset plays a crucial role in reducing computational costs and enhancing the effectiveness of SINDy-based algorithms. To this aim, we employ a greedy sampling approach to the snapshot matrix of a PDE to obtain its valuable samples, which are suitable to train a deep neural network (DNN) in a SINDy framework. SINDy based algorithms often consist of a data collection unit, constructing a dictionary of basis functions, computing the time derivative, and solving a sparse identification problem which ends to regularised least squares minimization. In this paper, we extend the results of a SINDy based deep learning model discovery (DeePyMoD) approach by integrating greedy sampling technique in its data collection unit and new sparsity promoting algorithms in the least squares minimization unit. In this regard we introduce the greedy sampling neural network in sparse identification of nonlinear partial differential equations (GN-SINDy) which blends a greedy sampling method, the DNN, and the SINDy algorithm. In the implementation phase, to show the effectiveness of GN-SINDy, we compare its results with DeePyMoD by using a Python package that is prepared for this purpose on numerous PDE discovery

Ali Forootani, Raffaele Iervolino

This paper proposes a unified family of learnable Koopman operator parameterizations that integrate linear dynamical systems theory with modern deep learning forecasting architectures. We introduce four learnable Koopman variants-scalar-gated, per-mode gated, MLP-shaped spectral mapping, and low-rank Koopman operators which generalize and interpolate between strictly stable Koopman operators and unconstrained linear latent dynamics. Our formulation enables explicit control over the spectrum, stability, and rank of the linear transition operator while retaining compatibility with expressive nonlinear backbones such as Patchtst, Autoformer, and Informer. We evaluate the proposed operators in a large-scale benchmark that also includes LSTM, DLinear, and simple diagonal State-Space Models (SSMs), as well as lightweight transformer variants. Experiments across multiple horizons and patch lengths show that learnable Koopman models provide a favorable bias-variance trade-off, improved conditioning, and more interpretable latent dynamics. We provide a full spectral analysis, including eigenvalue trajectories, stability envelopes, and learned spectral distributions. Our results demonstrate that learnable Koopman operators are effective, stable, and theoretically principled components for deep forecasting.

Ali Forootani, Mohammad Khosravi, Masoud Barati

Time series forecasting plays a vital role across scientific, industrial, and environmental domains, especially when dealing with high-dimensional and nonlinear systems. While Transformer-based models have recently achieved state-of-the-art performance in long-range forecasting, they often suffer from interpretability issues and instability in the presence of noise or dynamical uncertainty. In this work, we propose DeepKoopFormer, a principled forecasting framework that combines the representational power of Transformers with the theoretical rigor of Koopman operator theory. Our model features a modular encoder-propagator-decoder structure, where temporal dynamics are learned via a spectrally constrained, linear Koopman operator in a latent space. We impose structural guarantees-such as bounded spectral radius, Lyapunov based energy regularization, and orthogonal parameterization to ensure stability and interpretability. Comprehensive evaluations are conducted on both synthetic dynamical systems, real-world climate dataset (wind speed and surface pressure), financial time series (cryptocurrency), and electricity generation dataset using the Python package that is prepared for this purpose. Across all experiments, DeepKoopFormer consistently outperforms standard LSTM and baseline Transformer models in terms of accuracy, robustness to noise, and long-term forecasting stability. These results establish DeepKoopFormer as a flexible, interpretable, and robust framework for forecasting in high dimensional and dynamical settings.

Ali Forootani, Raffaele Iervolino

Federated Learning (FL) enables collaborative model training across decentralized devices while preserving data privacy. However, traditional FL suffers from communication overhead, system heterogeneity, and straggler effects. Asynchronous Federated Learning (AFL) addresses these by allowing clients to update independently, improving scalability and reducing synchronization delays. This paper extends AFL to handle non-convex objective functions and heterogeneous datasets, common in modern deep learning. We present a rigorous convergence analysis, deriving bounds on the expected gradient norm and studying the effects of staleness, variance, and heterogeneity. To mitigate stale updates, we introduce a staleness aware aggregation that prioritizes fresher updates and a dynamic learning rate schedule that adapts to client staleness and heterogeneity, improving stability and convergence. Our framework accommodates variations in computational power, data distribution, and communication delays, making it practical for real world applications. We also analyze the impact of client selection strategies-sampling with or without replacement-on variance and convergence. Implemented in PyTorch with Python's asyncio, our approach is validated through experiments demonstrating improved performance and scalability for asynchronous, heterogeneous, and non-convex FL scenarios.