A Dynamical Systems Perspective on the Analysis of Neural Networks (2507.05164v1)

Abstract: In this chapter, we utilize dynamical systems to analyze several aspects of machine learning algorithms. As an expository contribution we demonstrate how to re-formulate a wide variety of challenges from deep neural networks, (stochastic) gradient descent, and related topics into dynamical statements. We also tackle three concrete challenges. First, we consider the process of information propagation through a neural network, i.e., we study the input-output map for different architectures. We explain the universal embedding property for augmented neural ODEs representing arbitrary functions of given regularity, the classification of multilayer perceptrons and neural ODEs in terms of suitable function classes, and the memory-dependence in neural delay equations. Second, we consider the training aspect of neural networks dynamically. We describe a dynamical systems perspective on gradient descent and study stability for overdetermined problems. We then extend this analysis to the overparameterized setting and describe the edge of stability phenomenon, also in the context of possible explanations for implicit bias. For stochastic gradient descent, we present stability results for the overparameterized setting via Lyapunov exponents of interpolation solutions. Third, we explain several results regarding mean-field limits of neural networks. We describe a result that extends existing techniques to heterogeneous neural networks involving graph limits via digraph measures. This shows how large classes of neural networks naturally fall within the framework of Kuramoto-type models on graphs and their large-graph limits. Finally, we point out that similar strategies to use dynamics to study explainable and reliable AI can also be applied to settings such as generative models or fundamental issues in gradient training methods, such as backpropagation or vanishing/exploding gradients.

• The paper introduces a framework that reformulates neural network behaviors as dynamical systems problems, unifying analysis of architecture and training dynamics.
• It classifies networks into discrete and continuous forms, showing how memory and width augmentations critically enhance universal approximation and embedding.
• The study rigorously examines optimization stability of GD and SGD using Lyapunov exponents and mean-field limits, illuminating implicit bias and generalization.

A Dynamical Systems Perspective on the Analysis of Neural Networks

This work provides a comprehensive exposition of how dynamical systems theory can be leveraged to analyze and understand neural networks, both in terms of their architecture (information propagation) and their training dynamics (optimization). The authors systematically reformulate a wide range of neural network phenomena as dynamical systems problems, enabling the application of established mathematical tools from nonlinear dynamics, stochastic processes, and mean-field theory.

Neural Networks as Dynamical Systems

The paper distinguishes between two fundamental dynamical processes in neural networks:

• Dynamics on the Network: The propagation of information through a fixed network, formalized as the input-output map $\Phi_\theta(x)$ for fixed parameters $\theta$.
• Dynamics of the Network: The evolution of network parameters $\theta$ during training, typically via (stochastic) gradient-based optimization.

This dichotomy allows for a clear separation of concerns: the representational capacity of architectures can be studied independently from the properties of optimization algorithms.

Feed-Forward, Residual, and Continuous-Time Architectures

The authors provide a detailed classification of feed-forward neural networks (FNNs), including multilayer perceptrons (MLPs), ResNets, and DenseResNets, based on their layer width profiles (non-augmented, augmented, bottleneck). They then connect discrete architectures to their continuous-time analogs:

• Neural ODEs: Interpreted as the infinite-depth limit of ResNets, with the forward pass corresponding to the solution of an ODE.
• Neural DDEs: Generalize neural ODEs by incorporating memory via delays, corresponding to the infinite-depth limit of DenseResNets.

This continuous-time perspective enables the use of tools from the theory of ODEs and DDEs to analyze the representational and dynamical properties of deep networks.

Universal Approximation and Embedding

A central theme is the distinction between universal approximation (the ability to approximate any function in a given class arbitrarily well) and universal embedding (the ability to represent any function in the class exactly). The authors prove that:

• Augmented neural ODEs (with sufficiently large hidden dimension $m \geq d+q$) possess the universal embedding property for $C^k$ functions, a result that does not hold for non-augmented architectures.
• Non-augmented architectures (both MLPs and neural ODEs) are fundamentally limited in their representational capacity, as they cannot realize functions with arbitrary critical point structure.

The geometric analysis via Morse functions and the classification into function classes $(\mathcal{C}1)$, $(\mathcal{C}2)$, and $(\mathcal{C}3)$ provides a rigorous framework for understanding these limitations.

Memory and Universal Approximation

A novel contribution is the analysis of how memory capacity (quantified by the product of the Lipschitz constant $K$ and delay $\tau$ in neural DDEs) affects universal approximation. The authors establish sharp thresholds: for non-augmented neural DDEs, universal approximation is only possible if the memory capacity exceeds a certain bound, otherwise the class is strictly limited. This result formalizes the intuition that memory (via delay or skip connections) can compensate for limited width in deep architectures.

Optimization Dynamics: Stability and Implicit Bias

The training process is rigorously recast as a dynamical system in parameter space, with gradient descent (GD) and stochastic gradient descent (SGD) analyzed through the lens of stability theory.

Overdetermined vs. Overparameterized Regimes

• Overdetermined ($D < qN$): The loss landscape generically has isolated minima. The notion of Milnor stability is introduced to characterize the probability of convergence to a minimum under random initialization. The stability of a minimum is determined by the spectral radius of the Jacobian of the GD update map.
• Overparameterized ($D > qN$): The set of global minima forms a manifold of dimension $D - qN$. The stability of a minimum is governed by the transverse eigenvalues of the Jacobian restricted to the normal space of the manifold. The analysis reveals that GD preferentially converges to flatter minima, providing a dynamical explanation for implicit bias and the observed generalization properties of overparameterized networks.

The edge of stability phenomenon is discussed, where GD converges to minima at the threshold of stability, with the learning rate directly controlling the flatness of the selected minimum.

Stochastic Gradient Descent and Lyapunov Exponents

For SGD, the stability of minima is characterized by the top Lyapunov exponent of a random matrix product associated with the sequence of mini-batch updates. The authors prove that, for regular minima, local stability is equivalent to the Lyapunov exponent being negative. This result generalizes the deterministic stability condition to the stochastic setting and provides a rigorous foundation for understanding the convergence properties of SGD in high-dimensional, overparameterized regimes.

Mean-Field Limits and Network Heterogeneity

The paper extends the dynamical systems perspective to large-scale networks by considering mean-field limits of interacting particle systems (IPS). The authors show that:

• Many neural network architectures (including RNNs and transformers) can be viewed as special cases of IPS on graphs.
• In the large-width or large-network limit, the dynamics can be described by Vlasov-type PDEs or their generalizations, with rigorous convergence results under appropriate metrics (e.g., Wasserstein distance).
• Heterogeneous networks can be analyzed using graphon and digraph measure frameworks, enabling the paper of complex architectures beyond all-to-all coupling.

This approach unifies a broad class of models under a common mathematical framework and facilitates the transfer of results from kinetic theory and statistical mechanics to machine learning.

Broader Implications and Future Directions

The dynamical systems viewpoint advocated in this work has several important implications:

• Unified Analysis: It enables the application of a vast array of mathematical tools (bifurcation theory, Lyapunov stability, mean-field theory, stochastic processes) to the analysis of neural networks and their training algorithms.
• Architectural Insights: The results on universal approximation, memory, and critical point structure provide principled guidance for network design, particularly in the context of depth, width, and skip connections.
• Optimization and Generalization: The stability-based analysis of GD and SGD offers a theoretical explanation for empirical phenomena such as implicit bias, flat minima, and the edge of stability, with direct implications for hyperparameter selection and training strategies.
• Generative Models and Beyond: The dynamical perspective extends naturally to generative models (e.g., GANs, diffusion models), recurrent architectures, and even the analysis of backpropagation and vanishing/exploding gradients.

The authors suggest that future developments in AI will continue to benefit from the systematic application of dynamical systems theory, particularly as models and training algorithms become increasingly complex and high-dimensional. The integration of computer-assisted proofs and rigorous numerical methods is highlighted as a promising direction for addressing challenges that are analytically intractable.

Summary Table: Key Theoretical Results

Aspect	Main Result/Insight	Practical Implication
Universal Embedding	Augmented neural ODEs with $m \geq d+q$ can represent any $C^k$ function exactly	Justifies use of width augmentation for expressivity
Memory in Neural DDEs	Universal approximation possible if memory capacity $K\tau$ exceeds threshold	Memory/skip connections can compensate for width
GD Stability (Overparam.)	Only minima with flat enough loss landscape (small Hessian norm) are stable	GD implicitly selects flat minima (better generalization)
SGD Stability	Local stability determined by top Lyapunov exponent of random matrix product	Provides criterion for convergence under SGD
Mean-Field Limits	Large networks converge to Vlasov-type PDEs under appropriate scaling and coupling	Enables analysis of infinite-width/depth limits

Concluding Remarks

This work demonstrates that a dynamical systems perspective is not only natural but also highly effective for the rigorous analysis of neural networks and their training algorithms. By systematically translating architectural and optimization questions into dynamical systems problems, the authors provide a unifying mathematical framework that yields both theoretical insights and practical guidance for the design and analysis of modern machine learning systems. The results presented have direct implications for network architecture, training dynamics, and the understanding of generalization, and they open avenues for further research at the intersection of dynamical systems and machine learning.