Скачать или смотреть single CPU learning FNN/FFNN demo - real time AI - ( a m b i e n t ) -

training sequences of a fully functional neural network that learns to fit a 1D mathematical curve through backpropagation. the learning is fully reset every cycle and the points are then randomized for new pattern training. since FNN in this case has a single hidden layer (instead of several as with DNN) the curve only likes to bend two or maybe three times before setting itself. interestingly; if the random pattern turns out too advanced, then the line prefers to place itself at the absolute average of the points instead.

the code syntax / setup creates a shallow single-layer perceptron as the function approximator, meaning the model structure is constrained to f(x) = σ(w₂ · σ(w₁x + b₁) + b₂), where σ denotes the sigmoid activation. due to this compound nonlinearity, the function behaves like a limited universal approximator, able to emulate smooth transitions but struggling with discontinuities or high-frequency components unless trained perfectly or extended with more hidden units.

each training epoch involves backpropagation with a squared loss metric, where the gradient of error E = ½(y_target - y_pred)² is minimized using partial derivatives through the sigmoid chain. what's fascinating is the emergent behavior of the network weights as they settle into basins of low error; they don't search linearly but move through a dynamic energy landscape influenced by all parameters simultaneously. this becomes evident when the network converges into local minima that are sometimes suboptimal yet stable.

the training input space is normalized between 0 and 1, but the target outputs may include high curvature or inflection points. mathematically this means we're attempting to fit functions such as y = a · sin(bx) + c · cos(dx) + ε with a limited non-linear basis. during cases where the amplitude or frequency becomes too aggressive for the shallow network to replicate, the curve often regresses to its mean, a statistical midpoint minimization of loss rather than true pattern capture.

hidden layer outputs act as learned basis functions. with sigmoid activations, each hidden unit can be seen as a soft step-function or bump curve centered around a specific region in the input domain. by summing these outputs with different weights, the final curve can approximate composite shapes like f(x) = Σᵢ₌₁ᴺ wᵢ · σ(aᵢx + bᵢ). the more hidden nodes used, the richer the function space becomes, enabling tighter fits to chaotic patterns as long as overfitting is controlled.

the step-function and piecewise patterns reveal a lot about the limitations of sigmoidal compression. when approximating a jump between flat regions, the model produces an s-curve transition rather than a hard switch. this is a side-effect of the sigmoid's inability to express high gradients without saturation. the result is that instead of sharp corners, we see a smooth interpolation that under- or overshoots the correct values in the attempt to compromise between multiple distinct target zones.

by increasing the number of training points, we observe how the network's generalization capability becomes overwhelmed unless the model complexity scales proportionally. this aligns with the approximation theory trade-off: low-bias, high-variance models can memorize better but generalize worse. with fixed hidden nodes, increasing training resolution forces the curve to flatten or snap toward average clusters rather than honoring high-fidelity variations, a visible expression of the bias-variance dilemma.

so in visual terms, the curve fitting behavior can be treated as a form of emergent spline generation constrained by sigmoid basis smoothness. rather than cubic or bézier splines with explicit control points, the shape arises from distributed micro-adjustments to weights. the difference is that while spline functions are deterministic and geometric, neural ones are adaptive and probabilistic. each retraining session is a stochastic attempt at curve reconstruction using the same function class but newly randomized initial conditions.

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