What is a radial basis function network?

by Stephen M. Walker II, Co-Founder / CEO

What is a radial basis function network?

A Radial Basis Function Network (RBFN) is a type of artificial neural network that uses radial basis functions as activation functions. It's primarily used for function approximation, time series prediction, classification, and system control.

The architecture of an RBFN typically consists of three layers: an input layer, a hidden layer with a non-linear RBF activation function, and a linear output layer. The input layer receives the input data and feeds it into the hidden layer. The hidden layer then transforms the input, which might not be linearly separable, into a higher dimensionality based on Cover's theorem on the separability of patterns. The output layer performs the prediction task such as classification or regression.

RBFNs are unique in that they have a fundamentally different architecture than most neural network architectures, which typically consist of many layers and introduce nonlinearity by repetitively applying a non-linear function. Despite having only one hidden layer, RBFNs are proven to be universal approximators.

RBFNs have many applications, including stock price prediction, anomaly detection in data, fraud detection in financial transactions, and more. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment.

How does a radial basis function network work?

A Radial Basis Function Network (RBFN) operates by transforming the input data into a higher-dimensional space, where the patterns that might not be linearly separable in the original space can become separable. This transformation is achieved through the use of radial basis functions as activation functions in the hidden layer of the network.

Input Layer

The input layer of an RBFN consists of neurons that correspond to the dimensionality of the input data. These neurons are fully connected to the hidden layer but do not perform any computation themselves; they simply pass the input data to the hidden layer.

Hidden Layer

The hidden layer is where the transformation of the input data occurs. Each neuron in the hidden layer represents a center in the transformed space. The radial basis function, often a Gaussian function, is applied to the Euclidean distance between the input vector and the center vector associated with each hidden neuron. This process effectively measures how close or far the input data is from the center represented by each neuron. The output of each hidden neuron is a scalar value that represents this closeness.

Output Layer

The output layer is typically a linear layer that combines the activations from the hidden layer to produce the final output of the network. The output is a scalar function of the input vector, which can be used for tasks such as regression or classification. The output is computed as a weighted sum of the hidden layer activations.

What are the advantages and disadvantges of using a radial basis function network?

Radial Basis Function Networks (RBFNs) are a type of artificial neural network that have unique advantages and disadvantages.


  1. Easy Design — RBFNs have a simple structure with only one hidden layer, making them easier to design and understand.
  2. Good Generalization — RBFNs have a strong ability to generalize from the training data to unseen data.
  3. Faster Training — Due to their simple structure, RBFNs can be trained faster than other types of neural networks.
  4. Strong Tolerance to Input Noise — RBFNs are robust to noise in the input data, making them suitable for real-world applications where data may be imperfect or noisy.
  5. Interpretability — Each node in the hidden layer of an RBFN has a clear interpretation, which can aid in understanding the model's behavior.
  6. Universal Approximation — RBFNs are capable of approximating any continuous function to a desired degree of accuracy.


  1. Coverage of Input Space — RBFNs require good coverage of the input space by radial basis functions. The centers of these functions are determined by the distribution of the input data, but not with reference to the prediction task, which can lead to suboptimal performance.
  2. Slow Classification — Although RBFNs can be trained quickly, the classification process can be slower compared to other types of neural networks like Multi-Layer Perceptrons (MLPs).
  3. Limited Flexibility — RBFNs have only one hidden layer, which can limit their flexibility compared to other types of neural networks that have multiple hidden layers.

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