Auditory sensory overload affects 50-70% of individuals with Autism Spectrum Disorder (ASD), yet existing approaches, such as mechanistic models (Hodgkin Huxley type, Wilson Cowan, excitation inhibition balance), clinical tools (EEG/MEG, Sensory Profile scales), and ML methods (Neural ODEs, predictive coding), either assume fixed parameters or lack interpretability, missing autism heterogeneity. We present a Scientific Machine Learning approach using Universal Differential Equations (UDEs) to model sensory adaptation dynamics in autism. Our framework combines ordinary differential equations grounded in biophysics with neural networks to capture both mechanistic understanding and individual variability. We demonstrate that UDEs achieve a 90.8% improvement over pure Neural ODEs while using 73.5% fewer parameters. The model successfully recovers physiological parameters within the 2% error and provides a quantitative risk assessment for sensory overload, predicting 17.2% risk for pulse stimuli with specific temporal patterns. This framework establishes foundations for personalized, evidence-based interventions in autism, with direct applications to wearable technology and clinical practice.

We study learning of continuous-time inventory dynamics under stochastic demand and quantify when structure helps or hurts forecasting of the bullwhip effect. BULL-ODE compares a fully learned Neural ODE (NODE) that models the entire right-hand side against a physics-informed Universal Differential Equation (UDE) that preserves conservation and order-up-to structure while learning a small residual policy term. Classical supply chain models explain the bullwhip through control/forecasting choices and information sharing, while recent physics-informed and neural differential equation methods blend domain constraints with learned components. It is unclear whether structural bias helps or hinders forecasting under different demand regimes. We address this by using a single-echelon testbed with three demand regimes - AR(1) (autocorrelated), i.i.d. Gaussian, and heavy-tailed lognormal. Training is done on varying fractions of each trajectory, followed by evaluation of multi-step forecasts for inventory I, order rate O, and demand D. Across the structured regimes, UDE consistently generalizes better: with 90% of the training horizon, inventory RMSE drops from 4.92 (NODE) to 0.26 (UDE) under AR(1) and from 5.96 to 0.95 under Gaussian demand. Under heavy-tailed lognormal shocks, the flexibility of NODE is better. These trends persist as train18 ing data shrinks, with NODE exhibiting phase drift in extrapolation while UDE remains stable but underreacts to rare spikes. Our results provide concrete guidance: enforce structure when noise is light-tailed or temporally correlated; relax structure when extreme events dominate. Beyond inventory control, the results offer guidance for hybrid modeling in scientific and engineering systems: enforce known structure when conservation laws and modest noise dominate, and relax structure to capture extremes in settings where rare events drive dynamics.

Universal Differential Equations (UDEs), which blend neural networks with physical differential equations, have emerged as a powerful framework for scientific machine learning (SciML), enabling data-efficient, interpretable, and physically consistent modeling. In the context of smart grid systems, modeling node-wise battery dynamics remains a challenge due to the stochasticity of solar input and variability in household load profiles. Traditional approaches often struggle with generalization and fail to capture unmodeled residual dynamics. This work proposes a UDE-based approach to learn node-specific battery evolution by embedding a neural residual into a physically inspired battery ODE. Synthetic yet realistic solar generation and load demand data are used to simulate battery dynamics over time. The neural component learns to model unobserved or stochastic corrections arising from heterogeneity in node demand and environmental conditions. Comprehensive experiments reveal that the trained UDE aligns closely with ground truth battery trajectories, exhibits smooth convergence behavior, and maintains stability in long-term forecasts. These findings affirm the viability of UDE-based SciML approaches for battery modeling in decentralized energy networks and suggest broader implications for real-time control and optimization in renewable-integrated smart grids.

In this study, we apply two pillars of Scientific Machine Learning: Neural Ordinary Differential Equations (Neural ODEs) and Universal Differential Equations (UDEs) to the Lotka Volterra Predator Prey Model, a fundamental ecological model describing the dynamic interactions between predator and prey populations. The Lotka-Volterra model is critical for understanding ecological dynamics, population control, and species interactions, as it is represented by a system of differential equations. In this work, we aim to uncover the underlying differential equations without prior knowledge of the system, relying solely on training data and neural networks. Using robust modeling in the Julia programming language, we demonstrate that both Neural ODEs and UDEs can be effectively utilized for prediction and forecasting of the Lotka-Volterra system. More importantly, we introduce the forecasting breakdown point: the time at which forecasting fails for both Neural ODEs and UDEs. We observe how UDEs outperform Neural ODEs by effectively recovering the underlying dynamics and achieving accurate forecasting with significantly less training data. Additionally, we introduce Gaussian noise of varying magnitudes (from mild to high) to simulate real-world data perturbations and show that UDEs exhibit superior robustness, effectively recovering the underlying dynamics even in the presence of noisy data, while Neural ODEs struggle with high levels of noise. Through extensive hyperparameter optimization, we offer insights into neural network architectures, activation functions, and optimizers that yield the best results. This study opens the door to applying Scientific Machine Learning frameworks for forecasting tasks across a wide range of ecological and scientific domains.

Highly-interconnected societies difficult to model the spread of infectious diseases such as COVID-19. Single-region SIR models fail to account for incoming forces of infection and expanding them to a large number of interacting regions involves many assumptions that do not hold in the real world. We propose using Universal Differential Equations (UDEs) to capture the influence of neighboring regions and improve the model's predictions in a combined SIR+UDE model. UDEs are differential equations totally or partially defined by a deep neural network (DNN). We include an additive term to the SIR equations composed by a DNN that learns the incoming force of infection from the other regions. The learning is performed using automatic differentiation and gradient descent to approach the change in the target system caused by the state of the neighboring regions. We compared the proposed model using a simulated COVID-19 outbreak against a single-region SIR and a fully data-driven model composed only of a DNN. The proposed UDE+SIR model generates predictions that capture the outbreak dynamic more accurately, but a decay in performance is observed at the last stages of the outbreak. The single-area SIR and the fully data-driven approach do not capture the proper dynamics accurately. Once the predictions were obtained, we employed the SINDy algorithm to substitute the DNN with a regression, removing the black box element of the model with no considerable increase in the error levels.

In the last decade, the scientific community has devolved its attention to the deployment of data-driven approaches in scientific research to provide accurate and reliable analysis of a plethora of phenomena. Most notably, Physics-informed Neural Networks and, more recently, Universal Differential Equations (UDEs) proved to be effective both in system integration and identification. However, there is a lack of an in-depth analysis of the proposed techniques. In this work, we make a contribution by testing the UDE framework in the context of Ordinary Differential Equations (ODEs) discovery. In our analysis, performed on two case studies, we highlight some of the issues arising when combining data-driven approaches and numerical solvers, and we investigate the importance of the data collection process. We believe that our analysis represents a significant contribution in investigating the capabilities and limitations of Physics-informed Machine Learning frameworks.

Networked dynamical systems are common throughout science in engineering; e.g., biological networks, reaction networks, power systems, and the like. For many such systems, nonlinearity drives populations of identical (or near-identical) units to exhibit a wide range of nontrivial behaviors, such as the emergence of coherent structures (e.g., waves and patterns) or otherwise notable dynamics (e.g., synchrony and chaos). In this work, we seek to infer (i) the intrinsic physics of a base unit of a population, (ii) the underlying graphical structure shared between units, and (iii) the coupling physics of a given networked dynamical system given observations of nodal states. These tasks are formulated around the notion of the Universal Differential Equation, whereby unknown dynamical systems can be approximated with neural networks, mathematical terms known a priori (albeit with unknown parameterizations), or combinations of the two. We demonstrate the value of these inference tasks by investigating not only future state predictions but also the inference of system behavior on varied network topologies. The effectiveness and utility of these methods is shown with their application to canonical networked nonlinear coupled oscillators.

In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating climate simulations by 15,000x, can be handled by training UDEs.