Convergence of Nonlinear Dynamics and Machine Learning in Control and Prediction

Bridging Nonlinear Dynamics and Machine Learning: A Unified Approach to Control and Prediction

Recent advancements in the fields of nonlinear systems, control theory, and machine learning have converged towards a unified approach that leverages the strengths of both domains. This synthesis is driving innovations in model accuracy, control performance, and real-time feasibility across a spectrum of applications, from medical devices to quantum computing and beyond.

Nonlinear Systems and Control Theory

The integration of advanced mathematical frameworks and computational techniques has significantly enhanced our ability to model and control nonlinear systems. Key developments include the application of nonlinear model reduction methods, such as autoencoders, to simplify complex systems without sacrificing accuracy. The use of lifting techniques within nonlinear model predictive control (NMPC) frameworks has also shown promise in improving control accuracy and reducing settling times, making real-time applications more feasible.

Koopman operator theory (KOT) has emerged as a powerful tool for linearizing complex nonlinear dynamics, particularly in the control of functional electrical stimulation (FES) for gait assistance. This approach not only enhances real-time feasibility but also opens the door to personalized medical applications. Furthermore, research into the effects of quantization on data-driven linear prediction and control has provided valuable insights into the robustness of Koopman-based methods, with implications for the design of autonomous systems.

In the realm of electrochemical systems, the development of a fully analytical, experimentally validated nonlinear model for proton exchange membrane fuel cells (PEMFC) represents a significant leap forward. This model serves as a foundational tool for the design of nonlinear controllers and observers, highlighting the importance of control-oriented modeling in advancing technology.

Machine Learning and Neural Networks

The fusion of machine learning techniques with traditional control and system identification methods is revolutionizing the field. Neural networks and deep learning models are increasingly used to approximate complex dynamics and operators, enhancing the performance, interpretability, and efficiency of model predictive control (MPC) and partial differential equations (PDEs) solutions. Innovations such as physics-informed neural networks and meta-learning frameworks are enabling rapid adaptation and generalization of system identification methods, pushing the boundaries of what's possible in real-time predictions and control.

Bayesian Neural Networks (BNNs) and uncertainty quantification are also seeing significant advancements, with a focus on efficient, scalable, and hardware-friendly implementations. The integration of BNNs with analog hardware, for example, leverages the inherent noise of devices for variational inference, promising substantial energy savings. These developments are not only improving the accuracy of uncertainty estimates but also enhancing the reliability of machine learning models in handling uncertain data.

Scientific Machine Learning

The application of Physics-Informed Neural Networks (PINNs) in scientific machine learning is addressing complex problems across various domains, from structural mechanics to disease forecasting. By blending data-driven techniques with physical laws, PINNs are enabling more accurate predictions and solutions to previously intractable problems. The field is also seeing advancements in the use of neural networks for solving PDEs, with novel architectures and training strategies improving convergence and accuracy.

Conclusion

The convergence of nonlinear systems, control theory, and machine learning is driving a paradigm shift in how we approach complex problems. By leveraging the strengths of each domain, researchers are developing more accurate, efficient, and adaptable models and control strategies. These advancements are not only advancing our theoretical understanding but also having a profound impact on practical applications, from medical devices to autonomous systems and beyond.

Noteworthy Papers

Leveraging time and parameters for nonlinear model reduction methods
Enhanced sampled-data model predictive control via nonlinear lifting
Koopman-Based Model Predictive Control of Functional Electrical Stimulation for Ankle Dorsiflexion and Plantarflexion Assistance
Koopman Meets Limited Bandwidth: Effect of Quantization on Data-Driven Linear Prediction and Control of Nonlinear Systems
Nonlinear Modeling of a PEM Fuel Cell System; a Practical Study with Experimental Validation
A posteriori error estimates for the Lindblad master equation
Imitation Learning of MPC with Neural Networks: Error Guarantees and Sparsification
Stochastic Process Learning via Operator Flow Matching
Orthogonal projection-based regularization for efficient model augmentation
Meta-Learning for Physically-Constrained Neural System Identification
Physics-Informed Latent Neural Operator for Real-time Predictions of Complex Physical Systems
ELM-DeepONets: Backpropagation-Free Training of Deep Operator Networks via Extreme Learning Machines
A 65 nm Bayesian Neural Network Accelerator
Analog Bayesian neural networks are insensitive to the shape of the weight distribution
Learning dynamical systems with hit-and-run random feature maps
Compact Bayesian Neural Networks via pruned MCMC sampling
Big Batch Bayesian Active Learning by Considering Predictive Probabilities
Can Bayesian Neural Networks Explicitly Model Input Uncertainty?
Scalable Bayesian Physics-Informed Kolmogorov-Arnold Networks
Advanced Displacement Magnitude Prediction in Multi-Material Architected Lattice Structure Beams Using Physics Informed Neural Network Architecture
Inverse Design of Optimal Stern Shape with Convolutional Neural Network-based Pressure Distribution
Low-Order Flow Reconstruction and Uncertainty Quantification in Disturbed Aerodynamics Using Sparse Pressure Measurements
Physics Informed Neural Networks for Learning the Horizon Size in Bond-Based Peridynamic Models
DeepVIVONet: Using deep neural operators to optimize sensor locations with application to vortex-induced vibrations
Probabilistic Skip Connections for Deterministic Uncertainty Quantification in Deep Neural Networks
Conditional Diffusion Model for Electrical Impedance Tomography
Mechanics and Design of Metastructured Auxetic Patches with Bio-inspired Materials
AlphaNet: Scaling Up Local Frame-based Atomistic Foundation Model
Dynami-CAL GraphNet: A Physics-Informed Graph Neural Network Conserving Linear and Angular Momentum for Dynamical Systems
An Adaptive Collocation Point Strategy For Physics Informed Neural Networks via the QR Discrete Empirical Interpolation Method
Deep Learning for Disease Outbreak Prediction: A Robust Early Warning Signal for Transcritical Bifurcations
PINN-FEM: A Hybrid Approach for Enforcing Dirichlet Boundary Conditions in Physics-Informed Neural Networks
Conformal mapping Coordinates Physics-Informed Neural Networks (CoCo-PINNs): learning neural networks for designing neutral inclusions
BIAN: A boundary-informed Alone Neural Network for solving PDE-constrained Inverse Problems
Physics-informed neural networks for phase-resolved data assimilation and prediction of nonlinear ocean waves
Physics-Informed Machine Learning for Microscale Drying of Plant-Based Foods: A Systematic Review of Computational Models and Experimental Insights
Physics-informed deep learning for infectious disease forecasting