Feedforward Neural Networks

Inverse Kinematics (IK) plays a critical role in robotic motion planning and control. The IK solutions of a robot manipulator could be done by conventional ways such as geometric, algebraic, or Jacobian methods, which have drawbacks. The Artificial Neural Networks (ANNs) have become a promising alternative for approximating IK solutions due to their generalization ability and computational efficiency. This approach basically trains only a few samples of the end effector that are recorded for the solution of the IK problem. However, a fundamental question remains: how many training samples are sufficient to achieve reliable and accurate IK predictions? This study investigates the mathematical framework of relating the size of training datasets and the accuracy of ANN-based IK solvers. Using an articulated robotic manipulator, we generate varying amounts of joint-position pairs to train feedforward neural networks and assess their accuracy, convergence, and generalization capability. The results reveal more training samples than 125 did not contribute to the improvement of the model efficiency that the comparable measure dealing with the approximation accuracy over the sampling size, offering valuable insight into data efficiency. This work provides practical guidance for optimizing the data sizing of ANN solutions, balancing computational cost and model accuracy for real-world robotic applications.

We consider one-hidden layer neural networks trained in the feature-learning regime using gradient descent, and relate the output of the finite-width network $f_{\hatρ_t^m}$ to its infinite-width counterpart $f_{ρ_t^{MF}}$, which evolves in the mean-field dynamics. While constant-time horizon bounds for $\|f_{ρ_t^{MF}} - f_{\hatρ_t^m}\|$ may be obtained via standard Grönwall estimates, the long-time behavior of the fluctuation is a more delicate matter. Uniform-in-time bounds often rely on (local) strong convexity in the landscape or Logarithmic Sobolev inequalities present in noisy gradient dynamics. In this work, we establish non-asymptotic weak propagation-of-chaos that holds uniformly in time, obtained by exploiting instead the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. Specifically, denoting by $L_t$ the mean-field excess MSE loss at time $t$ and $m$ the number of neurons, under standard regularity assumptions and the condition $\int_0^\infty L_t^{1/2} dt =O(\log d)$, we obtain the uniform in time bound $\|f_{ρ_t^{MF}}- f_{\hatρ_t^m}\|^2 \lesssim \text{poly}(d) m^{-\min(1,c/6)}$ whenever $L_t \lesssim t^{-c}$. Our result holds in a noiseless setting and does not make any assumptions on the geometry of the landscape near the optimum, and extends seamlessly to other forms of discretization, including finite number of samples and time discretization. A key takeaway of our result is that whenever the convergence rate of the mean-field, population-loss dynamics is faster than $t^{-2}$, we can attain a loss of $ε$ with only $\text{poly}(d/ε)$ neurons, training samples, and GD steps.

Universal approximation theorems provide a mathematical explanation for the expressive power of neural networks. They assert that, under mild conditions on the activation function, feedforward neural networks are dense in broad function classes, such as continuous functions on compact subsets of $\mathbb{R}^d$, $L^p$ spaces, or Sobolev spaces. Over the past four decades, these qualitative universality results have evolved into a rich quantitative theory addressing approximation rates, parameter efficiency, and the role of architectural features such as depth and width. This survey presents several glimpses into this theory. We review classical density results for single-hidden-layer networks, as well as quantitative bounds that relate approximation error to network size and smoothness assumptions on target functions. Particular emphasis is placed on depth--width trade-offs and on results demonstrating that deeper architectures can achieve superior parameter efficiency for structured function classes. In addition to standard feedforward neural networks, we also review recent developments on Kolmogorov--Arnold Networks (KANs), which offer an alternative architectural paradigm and whose approximation-theoretic properties have begun to attract significant theoretical attention.

We show that, in a precise sense, a broad class of feedforward neural networks learn (have finite sample complexity) in the PAC model: every fixed finite feedforward architecture whose layers are definable in an o-minimal structure has finite sample complexity in the agnostic PAC setting, even with unbounded parameters. This covers standard fixed-size MLPs, CNNs, GNNs, and transformers with fixed sequence length, together with the operations and layers typically used in such architectures, including linear projections, residual connections, attention mechanisms, pooling layers, normalization layers, and admissible positional encodings. Hence, distribution-free learnability for modern non-recurrent architectures is not an exceptional property of particular activations or architecture-specific VC arguments, but a consequence of tame feedforward computation. Our results reposition finite-sample PAC learnability as a baseline rather than a differentiator: they shift the focus of architectural comparison toward inductive biases, symmetries and geometric priors, scalability, and optimization behaviour.

Climate classification plays a vital role in agricultural planning, hydrological studies, and climate science. One of the most widely used systems for classifying global climate zones is the Köppen-Trewartha (KT) classification. However, the KT classification is fundamentally deterministic, offering discrete labels to spatial locations without accounting for uncertainties in classification. In this paper, we provide a framework for probabilistic modeling of climatic zones. We implement a feedforward artificial neural network (ANN) for classification, allowing for efficient, uncertainty-aware categorization of climatic regions, thereby offering a more nuanced understanding of transitional climate zones compared to traditional deterministic methods. We apply this method to the Sahara Desert region over the 30-year period of 1960 - 1989, using data at more than 400,000 space-time locations from the first 11 years to train our model. We assess the model's short- and long-term classification capabilities to evaluate its stability and accuracy over time. We also compare the probabilistic classification from our model with the traditional KT classification. In addition, we use fluctuation analysis methods to highlight the temporal evolution of climatic zones across the Sahara region and identify areas undergoing significant flux of probabilities of their climate classes, providing insights into broader trends in desertification.

Communication Dynamics Neural Networks (CDNNs) apply the circulant-spectral machinery of the Communication Dynamics framework to neural-network layer design. We introduce CDLinear, a block-circulant linear layer with block size B = 2l + 1 that uses 1/B the parameters of a dense layer with the same input and output dimensions. The construction gives an explicit Fourier-domain diagnostic for optimization: for mean-squared loss, the weight Hessian is diagonalized by the discrete Fourier transform, with eigenvalues determined directly by the Fourier spectrum of the input blocks. Under input pre-whitening, the population Hessian condition number is exactly 1, and the empirical condition number is bounded by 1 + O(sqrt(B/N)) for N samples. We implement CDLinear in pure NumPy with hand-derived backward passes and verify gradients by finite differences. On the 8x8 MNIST digits benchmark, across three random seeds, a CDLinear MLP with B = 4 reaches 97.50% +/- 0.23% test accuracy using 2,380 parameters, compared with 98.15% +/- 0.47% for a dense baseline using 8,970 parameters. This gives a 3.8x parameter reduction at a 0.65% accuracy cost. The CD-MLP's mean Hessian condition number is 1.9e4, about 310x smaller than the dense baseline's 5.9e6. We position CDLinear as a special case of structured matrix neural-network layers, with the main contributions being a closed-form Hessian-spectrum diagnostic, a principled discrete sequence of block multiplicities, and an explicit conditioning analysis. We also release a reference PyTorch implementation integrating CDLinear into a DeepSeek-V3-style mixture-of-experts transformer for future large-scale benchmarks.

Neural networks transform data through learned representations whose geometry affects separation, contraction, and generalization. Recent work studies this geometry using discrete curvature on neighborhood graphs, suggesting Ricci-flow-like behavior across layers. We develop a smooth operator-theoretic alternative for feedforward representation snapshots. Each feature cloud induces a Gaussian-kernel diffusion Markov operator, and transport, spectral, label-boundary, and local-scale observables are derived from this single object via Bakry-Emery $Γ$-calculus. In a balanced Gaussian class-conditional snapshot model with shared covariance, the population operator has closed-form class affinities, leakage, and coarse spectra, all controlled by pairwise regularized Mahalanobis separations $c_\varepsilon^{(a,b)}$. We also prove that the resulting operator observables vary smoothly under feature perturbations, while hard neighborhood-graph diagnostics can change discontinuously. Synthetic experiments validate the closed-form Gaussian bridge, while learned MNIST experiments show that the same operator observables track training, width, and perturbation stability. Together, these results give a stable operator-geometric framework for analyzing feedforward representation geometry.

I propose the \emph{Random Cloud} method, a training-free approach to neural architecture search that discovers minimal feedforward network topologies through stochastic exploration and progressive structural reduction. Unlike post-training pruning methods that require a full train-prune-retrain cycle, this method evaluates randomly initialized networks without backpropagation, progressively reduces their topology, and only trains the best minimal candidate at the end. I evaluate on 7 classification benchmarks against magnitude pruning and random pruning baselines. The Random Cloud matches or outperforms both baselines in 6 of 7 datasets, achieving statistically significant improvements on Sonar ($+4.9$pp accuracy, $p{=}0.017$ vs magnitude pruning) with 87% parameter reduction. Crucially, the method is faster than both pruning baselines in 4 of 5 datasets (0.67--0.94$\times$ the cost of full training), since it avoids training the full-size network entirely.

These lecture notes provide an introduction to the verification of neural networks from a theoretical perspective. We discuss feed-forward neural networks, recurrent neural networks, attention mechanisms, and transformers, together with specification languages and algorithmic verification techniques.

Two deep ReLU networks can have entirely different architectures and parameters, yet realize the same function. We provide a complete characterization of this nonuniqueness. This is effected by building a symbolic calculus for deep ReLU networks, equivalence and simplification of networks becoming derivation of formulae, in close parallel to Shannon's analysis of switching circuits through Boolean logic. Inspired by Shannon, who turned circuit synthesis into the manipulation of Boolean formulae by the axioms of Boolean algebra, we turn ReLU network identification into the derivation of Łukasiewicz formulae by the axioms of many-valued (MV) logic. Two non-degenerate ReLU networks realize the same function on the unit cube if and only if one is obtained from the other by finitely many applications of the MV axioms for integer weights and biases, the divisible MV axioms for rational ones, and the Riesz MV axioms for real ones. The MV logic axioms characterize all symmetries of ReLU networks, the single-layer ones, which for tanh networks are the only kind, and the deep ones, spanning three or more layers. Our framework consists of three steps, an extraction algorithm turning a network into a substitution graph, whose represented formula has the network's input-output map as its truth function, a completeness theorem, by which functionally equivalent formulae are interderivable, and a construction algorithm returning from graphs to networks. The substitution graph is layered, carrying at each node a formula in the variables of the layer feeding it, encodes the network uniquely, and induces a new normal form for MV logic, compositional rather than flat as in the literature, hence retaining the algebraic structure of the network, with three local operations--node rewrite, layer collapse, layer expansion--realizing every derivation.

We propose a computational tropical geometry framework for the symbolic analysis of neural networks with tropical activations. The number of linear regions of a neural network has been actively studied as a measure of the expressivity of a given architecture. To study these, we work in the setting of tropical geometry---a combinatorial and polyhedral variant of algebraic geometry---where there are known connections between tropical rational maps and feedforward neural networks. We expand this connection by developing concrete computational tools for studying the linear regions of neural networks. We present an algorithm, together with a proof of correctness, which computes the linear regions of a neural network as explicit unions of polyhedra. We further relate the computation of the number of linear regions of a tropical expression to the number of monomials that appear in it, and show how tropical expressions can often be pruned to remove redundant monomials. We introduce the Hoffman constant of a neural network's tropical expression, a geometric quantity that controls the distance from any point in the input space to the farthest linear region. We provide the open source Julia library TropicalNN.jl, which is built on top of the OSCAR computer algebra system and implements the algorithms mentioned above to analyze neural networks symbolically using their tropical representations. We present a set of proof-of-concept computational examples to demonstrate how our tropical geometric theory can be applied to reveal insights on the expressivity of a network architecture.