Method for transforming a radial basis function neural network into a fuzzy inference system
Transforming RBFNs into fuzzy inference systems addresses the explainability issue by creating membership functions, enabling explainable and resource-efficient predictions for safety-critical applications.
Patent Information
- Authority / Receiving Office
- WO · WO
- Patent Type
- Applications
- Current Assignee / Owner
- THALES SA
- Filing Date
- 2025-11-14
- Publication Date
- 2026-05-21
AI Technical Summary
Existing radial basis function neural networks (RBFNs) are difficult to explain due to their black box nature, making them unsuitable for safety-critical applications like avionics and autonomous vehicles, where understanding predictions is essential for certification.
A method to transform RBFNs into fuzzy inference systems by dividing the network into subnetworks, adding inference layers, and degrading activation functions to create membership functions, enabling explainable predictive behavior.
The transformed fuzzy inference system provides explainable predictions, requiring reduced resources and being suitable for embedded systems, facilitating certification and deployment in avionics and autonomous driving.
Smart Images
Figure EP2025083053_21052026_PF_FP_ABST
Abstract
Description
[0001] METHOD FOR TRANSFORMING A RADIALLY BASED NEURAL NETWORK INTO A FUZZY INFERENCE SYSTEM
[0002] TECHNICAL FIELD
[0003] The present invention relates generally to the field of machine learning and more particularly to that of explainable neural networks. It finds application in the field of embedded artificial intelligence and especially in avionics.
[0004] PREVIOUS STATE OF THE ART
[0005] Artificial neural networks are well known from the state of the art for their ability to approximate arbitrary non-linear prediction functions.
[0006] However, neural networks (and especially deep neural networks, i.e., those with a large number of hidden layers) behave like black boxes in the sense that it is very difficult to understand the prediction model based solely on the network's parameters. A neural network is said to be explainable if it is possible to understand its predictions based on its input variables (or characteristics). These explanations can take the form of inference relationships linking output values to input variables using logical operators such as AND, OR, NOT, THEN, etc.
[0007] The explainability of a neural network is a key characteristic in the field of embedded artificial intelligence (AI) (autonomous vehicles, avionics). It is essential to be able to understand and validate decisions made by such AI in order to meet the required safety and / or certification standards.
[0008] Radial Basis Function Networks (RBFNs) are a very simple class of neural networks. An RBFN consists of an input layer, a single hidden layer, and an output layer. The input layer of an RBFN is a unit layer that transfers each input variable to one or more neurons in the hidden layer. The activation functions of the neurons in the hidden layer of the RBFN are radial functions.Aggarwal titled “Neural Networks in Deep Learning”, 2. nde edition, Chapter 6, published in 2023.
[0009] We also know the state of the art of fuzzy inference systems or FIS (Fuzzy Inference Systems). A fuzzy inference system is classically defined by a list of inference relations operating on fuzzy sets.
[0010] First, we recall that a fuzzy set is defined by its membership function: an element x belongs to a fuzzy set A if 0 < A (x) < 1 where p A is a continuous function called the membership function associated with A, A (x then represents the degree of membership of % in A. The relation x IS A (x IS A), also written x = A, means that x belongs to a set A with the degree of membership p A (x.
[0011] Logical operators can be defined based on membership functions. For example, the logical AND operator can be translated into a product or a minimum of membership degrees, in other words:
[0012] x = A AND y = B is quantified by A (x). p B y) or min( A (x), B (y)) (1)
[0013] Similarly, the logical OR operator can be translated into a maximum of degrees of membership, in other words:
[0014] x = A OR y = B is quantified by max(p A (x), p B (y)) (2)
[0015] Finally, the NOT operator is translated by a complement to 1 of the degree of membership:
[0016] NOT (% = A) is quantified by -BA(. X) (3) A fuzzy rule is expressed in the form of an IF (“situation”) THEN (“conclusion”) inference relation where the “situation”, also called the premise (antecedent) of the rule, is expressed in the form of a combination of membership functions relating to input variables and the “conclusion”, also called the consequence, relates to an output variable.
[0017] Fuzzy rules allow operation on "adjectives," each adjective corresponding to a fuzzy set. For example, an inference rule controlling the braking of an autonomous vehicle could be written as: Distance = "Low" AND Speed = "raised" THEN Braking = "Strong," where "Low," "High," and "Strong" are adjectives corresponding respectively to fuzzy sets in the space of distances, speeds, and pressure in the braking circuit.
[0018] Fuzzy rules can be of different types. In the FIS model following the Takagi-Sugeno model, the conclusion is crisp. Thus, for example, the conclusion is... 1 - of rule i for the output variable y7, j =
[0019]
[0020] can be expressed as a linear relationship of the input variables x k , k =
[0021]
[0022]
[0023] where the A l k are fuzzy sets.
[0024] In contrast, in a so-called Mamdani model, the conclusion yj of the same rule i for the output variable y,- is expressed in the form of a degree of membership of the latter in a fuzzy set:
[0025] IF (X1= AND X2= AND... ANDx p = ) THEN yj = Ç for j = l,..,q (5) where the A l kand the Cj are fuzzy sets which respectively define a partitioning of the input variable space and a partitioning of the output variable space.
[0026] It was shown in the article by J.-SR Jang and C.-T. Sun entitled "Functional equivalence between Radial Basis Function Networks and Fuzzy Inference Systems" published in IEEE Trans, on Neural Networks, vol. 4, no. 1, pp. 156-159, as well as in Chap. 12 of the book by J.-SR Jang, entitled "Neuro-Fuzzy and Soft Computing: a computational approach to learning and machine intelligence", 1st edition, 1997, that an RBFN network could be considered as functionally equivalent to a class of fuzzy inference systems, called "Sugeno ANFIS" or more simply "ANFIS" (Adaptive Network Fuzzy Inference Systems), obeying the aforementioned Takagi-Sugeno model.ANFIS fuzzy inference systems use hybrid learning: backpropagation learning to determine the parameters of membership functions and Least Mean Squares (LMS) estimation to determine the parameters of linear relationships (such as (4)) appearing in the conclusions of the Takagi-Sugeno inference model. However, these piecewise linear relationships do not provide a satisfactory explanation of the predictive behavior of the RBFN network.
[0027] The article by Y. Chai et al. entitled "Mamdani model based adaptive neural fuzzy inference system and its application," published in the International Journal of Computer and Information Engineering, 2009, vol. 3, no. 3, pp. 663-670, proposes a class of neural networks equivalent to a fuzzy inference system based on a Mamdani model. Such a neural network, called "M-ANFIS," has the advantage, by design, of being explainable, since the inference rules of the Mamdani model link adjectives using logical operators. However, training such an M-ANFIS network is complex and presents risks of divergence and overfitting.
[0028] The aim of the present invention is to propose a method for transforming a radially based neural network, whose input variables are representative of physical measurements of an embedded system, into a fuzzy inference system functionally equivalent to this neural network and explaining the predictive behavior of said network.
[0029] To this end, the invention relates to a method for transforming a radial basis neural network (RBFN), whose input variables represent physical measurements, into a fuzzy inference system. The RBFN comprises an input layer of N input nodes, each input node receiving one of said input variables, a single intermediate layer of H neurons, and an output layer of S output nodes, each output node providing an output variable. N, H, and S are integers greater than or equal to 1. Each neuron in the intermediate layer is characterized by a radial activation function centered on a central point c. h and of radius r h , h =
[0030]
[0031] said RBFN network being represented by a directed graph, said method being original in that it comprises:
[0032] - a step of dividing the RBFN network into S distinct RBFN subnetworks, each RBFN subnetwork being associated with a distinct output variable and being defined by an induced subgraph of said directed graph, retaining from the output layer only the node providing said output variable;
[0033] - a step of separating each of the S subnets to obtain S separate subnets, each separate subnet being formed by the set of distinct paths of that subnet between the input nodes and the output node providing the output variable associated with the subnet;
[0034] - for each subnetwork thus separated, a step of adding a layer of inference nodes at the output of the intermediate layer, each inference node applying an inference rule, the outputs of the inference nodes being combined to provide a value of the output variable associated with this subnetwork;
[0035] - training each subnetwork thus obtained on a training dataset;
[0036] - a degradation of the activation functions of the neurons in the intermediate layer, for each of the S separate subnetworks thus trained, the activation functions of the neurons in the intermediate layer connected to a node of the input layer being degraded into membership functions representing a partitioning of the space of the input variable; - the generation of a fuzzy inference system from said membership functions and the previously adjoint inference rules.
[0037] According to other advantageous aspects of the invention, the neural network transformation method comprises one or more of the following features, taken individually or in any technically possible combination:
[0038] The activation functions of the neurons in the intermediate layer can be Gaussian functions, symmetrical triangular functions, or trapezoidal-shaped functions.
[0039] Preferably, each inference node applies an inference rule based on the outputs of two neurons in the intermediate layer.
[0040] According to an advantageous embodiment of the invention, the degradation of the activation functions F q , q = 0, .. , k of the intermediate layer neurons connected to the same input node is obtained by selecting the centers of the radial activation functions of these neurons, c q , q = 0,..,k, and by constructing a set of membership functions of triangular shape, two membership functions of arbitrary consecutive centers of this set exhibiting an overlap.
[0041] These membership functions can then be obtained by means of:
[0042] F0(x) = min(l,max(0, x));
[0043] F q (%) = max(0, min fq~ (%), f q (%)) for q = 1, .. , k — 1;
[0044]
[0045] Each inference rule associated with an inference node can be expressed by means of said membership functions.
[0046] Alternatively, each inference rule involves the product of two membership functions associated with distinct neurons.
[0047] A subsidiary objective of the present invention is to be able to replace a radially based neural network, implemented in an embedded system, with a functionally equivalent fuzzy inference system but requiring only reduced resources in terms of memory and computing power.
[0048] To this end, each inference rule can be expressed as a multivariate polynomial whose variables are input variables of the RBFN network. This method includes a step of calculating the coefficients of the multivariate polynomial from the centers of the radial activation functions of the RBFN network. The coefficients of the multivariate polynomials expressing the inference rules of the fuzzy inference system, as well as the centers of the radial activation functions of the RBFN network, can then be simply stored in memory.
[0049] In any case, the input variables of the BFN network can be physical measurements provided by sensors of an embedded system and the output variables can be control parameters of said system.
[0050] BRIEF DESCRIPTION OF THE DRAWINGS
[0051] The invention will become clearer upon reading the following description, given solely by way of non-limiting example, and made with reference to the drawings in which:
[0052] - Figure 1 schematically illustrates a method for transforming a radially based neural network into a fuzzy inference system according to a general embodiment of the present invention;
[0053] - Figure 2 illustrates an example of a radially based neural network;
[0054] - Figure 3 illustrates the result of the processing of the neural network of Figure 2 at the end of the division step of Figure 1;
[0055] - Figure 4 illustrates the result of the processing of the neural network of Figure 2 at the end of the separation step of Figure 1;
[0056] - Figure 5 illustrates the result of the processing of the neural network of Figure 2 at the end of the step of inserting a layer of inference nodes of Figure 1;
[0057] - Figure 6 illustrates a process of obtaining membership functions from neuronal activation functions, as implemented in the activation function degradation step of Figure 1;
[0058] - Figure 7 illustrates the result of the processing of the neural network of Figure 2 at the end of the degradation step of the activation functions of Figure 1;
[0059] - Figure 8 schematically illustrates a method of transforming a radially based neural network into a fuzzy inference system according to a particular embodiment of the present invention.
[0060] DETAILED EXPLANATION OF THE METHODS OF IMPLEMENTATION
[0061] We will subsequently consider a radially based artificial neural network (RBFN), as defined above. This RBFN receives as input variables representing physical measurements, typically physical measurements provided by sensors of an embedded system. After training on a training dataset, the RBFN is able to provide output variables, for example, control parameters of the embedded system, in the form of control signals for units of that system. The input variables are represented hereafter by a vector x of size N, and the output variables by a vector y of size S.
[0062] Such an RBFN network is schematically represented in Figure 2. It comprises an input layer, 210, composed of N input nodes, here 211-213, each input node of the input layer receiving an input variable, and an intermediate layer, 220, composed of H neurons, here 221-224, each neuron h =
[0063]
[0064] of the intermediate layer being characterized by a radial activation function F hand finally an output layer, 230, composed of S output nodes, here 231-232. The connections between the input nodes and the neurons of the intermediate layer can be represented by an adjacency matrix M of size H x N, in which M(h,ri) = 1 if there is a connection between the input n of the input layer and the neuron h of the intermediate layer, and M(h,ri) = 0 otherwise. Without loss of generality, we can assume that each output node of the output layer is connected to all the neurons of the intermediate layer.
[0065] If we denote W in the H x N size matrix of weights between the input layer and the intermediate layer and W 0llt The S x H size matrix of weights between the intermediate layer and the output layer has the following relationships:
[0066] z in = (MQW in )x (6-1) y = w ou t Zout (6-2) with z 0llt (h) = F h
[0067]
[0068] where F h is a radial activation function centered at c h and of radius r h , the radius r h giving the extension of the support of this function around the center c h We define the vectors c and r of size H respectively by c = (c h ) h =
[0069]
[0070] and r =
[0071]
[0072] The advantage of an RBFN network is its very simple architecture and the relatively easy learning of it via backpropagation of the gradient. It is also a universal approximator in the sense that it can approximate any prediction function. After training, the RBFN network is uniquely characterized by the parameters N, H, S, M, W ln , W 0llt ,c,r , since the general form of the activation functions F h is known. For example, the functions Fh can be Gaussians respectively centered on the centers c h and standard deviations r h .
[0073] Such an RBFN network can be transformed into a fuzzy inference system using the transformation method according to a general embodiment of the invention described below in relation to Figure 1. In a first step, 110, the RBFN network is divided into S distinct RBFN subnetworks, each of these S subnetworks being associated with an output variable s = 1, ..., ,,S of the RBFN network. The RBFN subnetwork associated with the output s is obtained as an induced subgraph of the directed graph representing the original RBFN network, in which only the node associated with the output variable s is retained in the output layer. The RBFN subnetwork associated with the output variable s simply provides the s-th component of the vector y.
[0074] Figure 3 schematically illustrates the result of the splitting step for the RBFN network of Figure 2. The two subnetworks 241 and 242 correspond respectively to the subgraphs induced by keeping respectively in the output layer of the original RBFN graph the output nodes 231 and 232.
[0075] In a second step, 120, each of the S RBFN subnetworks undergoes a path separation operation. This separation step consists, for a given RBFN subnetwork, of constructing a directed graph, called the separated RBFN subnetwork, equivalent to the graph of the RBFN subnetwork in question, but in which the paths between the input nodes and the output node do not share any nodes in the intermediate layer. In other words, the separated subnetwork is obtained by instantiating each neuron in the original intermediate layer, the number of instances of that neuron being equal to its degree entering the RBFN subnetwork. The number of neurons in the resulting intermediate layer is then equal to:
[0076]
[0077] where deg h is the incoming degree of neuron h in the original RBFN network.
[0078] The inward degree of the neurons in the middle layer of each separate RBFN subnetwork is 1. In other words,
[0079]
[0080] " = 1 where M is the H x N size adjacency matrix of the separated RBFN subnetwork and h denotes the index of a neuron in the intermediate layer of the latter.
[0081] Figure 4 schematically illustrates the result of the separation step applied to the RBFN 241 subnetwork of Figure 3. It can be verified that the paths between the input nodes 211, 212, and 213 and the output node 223 are indeed separated at the intermediate layer. To achieve this, neuron 221 of the original intermediate layer was instantiated twice (221, 221'), neuron 222 was instantiated twice (222, 22'), while neuron 223 was instantiated only once (223), and finally, neuron 224 was instantiated twice (224 and 224').
[0082] It should be noted that since the RBFN subnets 241 and 242 have an identical structure, the separation of subnet 242 would lead to a result identical to that illustrated in figure 4.
[0083] In a third step, 130, called the inference rule insertion step, a layer of inference nodes, 250, is inserted for each separated RBFN subnetwork between the intermediate layer and the output node of that subnetwork. More precisely, each inference node i is associated one-to-one with an inference rule 91, combining all or some of the outputs of the neurons in the intermediate layer. The activation functions of the different neurons in the separated RBFN subnetwork can be considered as providing degrees of membership of each of the input variables in fuzzy sets.
[0084] Inference rules 91 can result from empirical knowledge, hypotheses, or expertise in the application domain under consideration. They allow us to identify dependency relationships between the output variable y and certain input variables, but are insufficient to explain the prediction of the output variable (to be specified / completed).
[0085] The inference rule 9Î, can for example be written in the following form:
[0086] IF (1= 1(0 AND x2= 2(0 AND... AND X N = A^) THEN y = / (8) where
[0087]
[0088] are fuzzy sets and p l is the result of inference rule 9î £ ,
[0089]
[0090] ..., hN(i) are the indices of the neurons in the intermediate layer of the separated subnetwork to which inference node i is connected. In equation (8), for the sake of notational simplification, it was assumed that each inference rule involved only one neuron per input value; in other words, that there was only one path between an input node and an inference node. However, those skilled in the art will understand that this constraint can generally be lifted.
[0091] In the example inference rule given in (8), all the logical operators are "AND". However, it is understood that some or even all of them may be logical operators of another type, based, for example, on conjunctions or disjunctions (e.g., exclusive "OR"). The inference rule
[0092]
[0093] may involve fewer than N neurons, in other words some input variables may not be taken into account in the rule in question.
[0094] The premise of inference rule 91, for example, is quantified by:
[0095]
[0096] where z 0llt is the vector of size H whose components are the outputs of the considered separate RBFN subnetwork and hn( ) the indices of the neurons of the intermediate layer taken into account in the inference rule 9Î,.
[0097] In general, the premise of the rule can relate to one, two, or N outputs of neurons from the intermediate layer.
[0098] Preferably, however, each inference rule combines only the outputs of two neurons in the intermediate layer (2-by-2 combination) connected to distinct input variables. Relation (9) then simply reduces to z out (hn(C)). z out (hn' (j)).
[0099] The results of the inference rules are then normalized, namely:
[0100]
[0101] where I is the number of inference rules (or equivalently the number of nodes)
[0102]
[0103] is a weighting coefficient.
[0104] For each separate RBFN subnetwork associated with an output variable s, the results of the different inference rules, after being normalized, are linearly combined using output weights. However, those skilled in the art will understand that these output weights cannot be identified with the elements of the s. ième row vector of the output matrix W out insofar as an inference layer has been introduced and the combination now no longer focuses on the outputs of the original H network neurons but on the I normalized outputs of the inference nodes.
[0105] To address this information loss, step 140 involves training each of the RBFN subnetworks obtained in step 130 for s = 1, ..., S using a training dataset. This training step employs a backpropagation algorithm based on a loss function, a method known per se. It allows us to obtain the output weights for each of the RBFN subnetworks. Figure 5 illustrates the result of processing the neural network from Figure 2 after the step of inserting an inference node layer, designated here as 250, into the separate RBFN subnetwork of Figure 4.
[0106] The number of inference rules here is I = 6. The inference rule
[0107]
[0108] relates to the outputs of neurons 221, 221', the inference rule 9Î2 relates to the outputs of neurons 221, 224 and 224',.., the inference rule 9Î6 relates to the outputs of neurons 222' and 224'.
[0109] Thus, by way of illustration, the premise of the inference rule
[0110]
[0111] is, for example, quantified by:
[0112] l))- 2) = F1(^ n (l,l).%1).F1(^ n (l,2).%2)
[0113] (11-1) and the premise of the inference rule 9Î6par:
[0114] z out (4).z out (6) = F3(^„(3,2).x2).F4(^„(4,3).x3)
[0115] (11-2)
[0116] Once the inference layers are inserted into the different separate RBFN subnetworks (associated with the respective outputs s = 1, ..., S of the original network) and these subnetworks are trained during the learning step 140, the activation functions of the neurons in each of these subnetworks are degraded in step 150. Indeed, to obtain a fuzzy inference system (FIS), it is necessary that, for a given input variable, the sum of the membership functions of this input variable in the different fuzzy sets constituting the input space be equal to 1; in other words, that the different fuzzy sets form a partition of the input variable's space. Note: this only appears to be true if the elements of the matrix Wj n i.e., the input weights, are all equal to 1; otherwise, the arguments of the different membership functions are different (to be verified).
[0117] For a given input variable x (the index n of the variable is omitted here for the sake of simplicity), we respectively denote F q , q = 0,..,k (with q = hn ) are the respective radial activation functions of the k + 1 neurons of the intermediate layer to which the input node corresponding to this variable is connected. These radial activation functions have their respective centers c q and for rays r q , i = 0, ,.,k. Without loss of generality, we will assume that the activation functions have been sorted by increasing values of their centers, that is, c q , > c q if q' > q.
[0118] From the centers c q , q = 0, .. , k we construct degraded activation functions, F q of the same centers as the radial functions F q but possessing an asymmetrical triangular shape. More precisely, we first define the increasing functions f q, q = l,..,k and dec
[0119]
[0120] The functions f~ and f q + cancel each other out respectively at c q-1 etc q+1 and both take the value 1 in c q .
[0121] Degraded activation functions F q , q = 0,..,k are then obtained as piecewise affine functions from the functions f~ and f q + :
[0122]
[0123] Ffc(x) = min(l, max(0, f k (x); (13-3)
[0124] Degraded activation functions F q , q = 0, ,.,k are overlapping on IR (with overlap of adjacent functions F q and F q+1 ) in the sense that, for any input value x of IR, there exists at least one value qe {0,..,k} such that F q (x) > 0.
[0125] They also verify the following property:
[0126]
[0127] They can therefore be considered as membership functions of the fuzzy sets j-oo,^], [c0, c2],.., [c k-2 ,c k ],..., [c k-1 ,+œ[.
[0128] Figure 6 illustrates, using an example, the degradation of activation functions into membership functions, as implemented in step 150 of Figure 1.
[0129] In the illustrated case, the radial activation functions F q , q = 0, ,.,k (with k = 3) are Gaussians centered on the values c q , q = 0,..,k. The degraded activation functions, F q , q = l,..,k- 1 are centered on the values c q , q = 1, ,.,k - 1 and the functions F o and F k , correspond to define the limits of the activation domain.
[0130] Given that the centers c q, q = 0,..,k activation functions are not necessarily equidistant, degraded activation functions F q q = 1, ..., k - 1 have any triangular form, asymmetric or not, the functions F o and F k being considered as special cases where one of the slopes is zero (step functions with a ramp). Figure 7 schematically illustrates the RBFN subnetwork of Figure 5 after step 150 of activation function degradation. The membership functions F are recognizable. q for each of the input variables. For example, the membership function F o relative to the input variable x1 is constant (a single activation function), the 4 membership functions F o F1, F2, F3 for the input variable x2 correspond to the centers of the 4 activation functions c q , q = 0,..,3, hence 2 step functions with ramp F o, F3 and 2 asymmetric triangular functions, Fi, F2, and finally the 2 membership functions F o , F for the input variable x3 are step functions with ramp.
[0131] After degradation of the activation functions according to membership, we have, for each RBFN subnetwork associated with an output variable, a set of inference rules relating to fuzzy sets (in other words, "adjectives" qualifying the different input variables and the corresponding output variable, therefore a Mamdani fuzzy inference system (FIS), natively explainable.
[0132] The resulting fuzzy inference system also has the advantage of being easily embedded, since its input variables can be represented by a simple polynomial. Indeed, when the membership functions p tof the output variable for the different rules 9Î, , i = 1, ... , I reduce to singletons (denoted below u , the output variable of the RBFN s = 1, ... , S is expressed, for example, in the form:
[0133] &
[0134]
[0135] where W out is a matrix of size S x / giving the output weights after the learning phase, W in is an H x N matrix giving the input weights of the separated RBFN subnetwork and
[0136]
[0137] the membership function involved in inference rule 9 £ , obtained after degradation of the activation functions of neurons connected to the input node n.
[0138] Given that each function
[0139]
[0140] , n = l,..,N is piecewise affine, the output variable y sis a piecewise multivariate polynomial function in x n , n = 1, ,.,N, that is to say of general form:
[0141] >
[0142]
[0143] where k), k n (i) are integers depending on the respective number of occurrences of the input variables x n , n = l,..,N in the inference rules
[0144]
[0145] i = 1, ..., I and coefficients are calculated from the centers of the activation functions
[0146]
[0147] radials of the RBFN network.
[0148] The storage of coefficients and centers of activation functions
[0149]
[0150] radial base c qq = 0, ..., k, that is, the bounds of the supports for degraded activation functions, meaning that, being inexpensive in memory and the calculation of the piecewise polynomial form (16) being very simple, this embodiment is particularly well suited to embedded applications, notably in an avionics system or an autonomous driving vehicle. Furthermore, the fuzzy inference system thus obtained, functionally equivalent to the original RBFN, is explainable to a person skilled in the art, which greatly facilitates its certification when required.
[0151] Figure 8 schematically illustrates a method for transforming a radially based neural network into a fuzzy inference system according to a particular embodiment of the present invention.
[0152] This method includes steps of dividing into RBFN subnetworks, 310, path separation in each induced subgraph, 320, insertion of an inference layer, 330, training on a training dataset, 340, degradation of the activation functions of the neurons in the intermediate layer, 350, and obtaining a fuzzy inference system, 360, respectively identical to steps 110, 120, 130, 140, 150, and 160 of Figure 1. For this reason, the description of the aforementioned steps will not be repeated here. This embodiment differs from that of Figure 1 in that it includes, in 370, a step for calculating the coefficients of the coefficients.
[0153]
[0154] of the piecewise multivariate polynomial, representative of the fuzzy inference system, the variables being the input variables of the original RBFN network, x n , n = l,..,N.
[0155] Finally, in step 380, the coefficients of the multivariate polynomials as well as the bounds, c q , q = 0,..,k, supports of piecewise multivariate polynomials degraded by the neurons of the intermediate layer are stored in an embedded memory. As previously indicated, it should be recalled that the bounds c q , q = 0, . ,,k, are none other than the centers of the radial activation functions of the original RBFN network.
[0156] It is therefore possible to substitute for an RBFN neural network a fuzzy, lightweight and explainable inference system, which is functionally equivalent and can be easily embedded.
Claims
DEMANDS 1. Method for transforming a radial basis neural network (RBFN), whose input variables represent physical measurements, into a fuzzy inference system. The RBFN comprises an input layer of N input nodes, each input node receiving one of the input variables, a single intermediate layer of H neurons, and an output layer of S output nodes, each output node providing an output variable. N, H, and S are integers greater than or equal to 1. Each neuron in the intermediate layer is characterized by a radial activation function centered on a center c. h and of radius r h , h = said RBFN network being represented by a directed graph, said method being characterized in that it comprises: - a division step (110, 310) of the RBFN network into S distinct RBFN subnetworks, each RBFN subnetwork being associated with a distinct output variable and being defined by an induced subgraph of said directed graph, retaining from the output layer only the node providing said output variable; - a separation step (120, 320) of each of the S subnets to obtain S separate subnets, each separate subnet being formed by the set of distinct paths of that subnet between the input nodes and the output node providing the output variable associated with the subnet, and the paths between the input nodes and the output node not sharing any node of the intermediate layer; - for each subnetwork thus separated, a step of adding a layer of inference nodes (130, 330) at the output of the intermediate layer, each inference node applying an inference rule, the outputs of the inference nodes being combined to provide a value of the output variable associated with this subnetwork; - training (140, 340) of each subnetwork thus obtained on a training dataset; - a degradation (150, 350) of the activation functions of the neurons in the intermediate layer, for each of the S separate subnetworks thus trained, the activation functions of the neurons in the intermediate layer connected to a node of the input layer being degraded into membership functions representing a partitioning of the input variable space; the degradation of the activation functions F q, q = 0, .. , k of the intermediate layer neurons connected to the same input node being obtained by selecting the centers of the radial activation functions of these neurons, c q , q = 0,..,k, and by constructing a set of membership functions of triangular shape, two membership functions with any consecutive centers of this set exhibiting an overlap; each inference rule associated with an inference node being expressed by means of said membership functions, and each inference rule being expressed in the form of a piecewise multivariate polynomial whose variables are input variables of the RBFN network, said method comprising a calculation step (370) of the coefficients of the multivariate polynomial from the centers of the radial activation functions of the RBFN network; - the generation of a fuzzy inference system (160, 360) from said membership functions and previously adjoined inference rules, the fuzzy inference system being implemented in an embedded system including sensors, the input variables being physical measurements provided by the sensors of the embedded system and the output variables being control parameters of said system.
2. Method according to claim 1, characterized in that the activation functions of the neurons in the intermediate layer are Gaussian functions, symmetric triangular functions or trapezoidal shape functions.
3. Method according to claim 1 or 2, characterized in that each inference node applies an inference rule relating to the outputs of two neurons of the intermediate layer.
4. A method according to any one of the preceding claims, characterized in that said membership functions are obtained by means of:
5. Method according to any one of the preceding claims, characterized in that each inference rule involves the product of two membership functions associated with distinct neurons.
6. Method according to any one of the preceding claims, characterized in that it comprises a memory storage (380) of the coefficients of the piecewise multivariate polynomials expressing the inference rules of the fuzzy inference system as well as the centers of the radial activation functions of the RBFN network. 17 7. Embedded system comprising sensors and implementing a fuzzy inference system, characterized in that the fuzzy inference system is generated by a method according to one of the preceding claims, the input variables being physical measurements provided by the sensors of the embedded system and the output variables being control parameters of said system, in the form of control signals of units of this system.