تناولت العديد من البحوث تقريب الدوال المستمرة باستخدام الشبكات العصبية. في هذا البحث استخدمنا الشبكات العصبية لتقريب الدوال في الفضاء &nbsp;عندما . اذ قمنا بدراسة تقريب الدوال المعرفة على الخط الحقيقي الكامل باستخدام الشبكات العصبية التي لها وزن ثابت. كما استخدمنا الالتفاف لتقريب الدوال في الفضاء &nbsp;عندما &nbsp;معرفة على فترة مرصوصة باستخدام الشبكات العصبية التي لها اوزان ثابتة.&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Many papers introduced about the approximation of continuous functions using neural networks. In this paper we use neural network to approximate functions in &nbsp;spaces for p . We study the approximation for functions defined on the complete real line using neural networks with radial basis having constant weight. We also use convolution to approximate functions in &nbsp;spaces for a compact interval using radial basis function neural networks of constant weights

Bhaya, Eman Samir

Sharba, Zainab Abdulmunim

English

Journals of University of Babylon

Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019   © Journal of University of Babylon for Pure and Applied Sciences (JUBPAS) by University of Babylon is licensed under a Creative Commons Attribution 4. 0 International License   400  Approximation of Functions in 𝐋𝐏 Spaces for 𝐩 < 𝟏,Using Radial Basis Function Neural Networks Eman Samir Bhaya Mathematics Department, College of Education for Pure Sciences, University of Babylon. emanbhaya@itnet.uobabylon.edu.iq Zainab Abdulmunim Sharba Computer Science Department, College of Science for Women, University of Babylon. zainb_sharba@yahoo.com    Abstract    Many papers introduced about the approximation of continuous functions using neural networks. In this paper we use neural network to approximate functions in Lp spaces for p < 1. We study the approximation for functions defined on the complete real line using neural networks with radial basis having constant weight. We also use convolution to approximate functions in Lp spaces for 𝑝 < 1 on a compact interval using radial basis function neural networks of constant weights.  Keywords: 𝐿𝑝 spaces, constructive approximation, neural networks, convolution.  1. Introduction    Many papers (see [1],[2],[3]and[4]) showed that we can consider the multilayered neural networks with radial basis function, as a universal approximation of a continuous function. Then in [5] the authors proved theorems for the approximation of functions in 𝐶2 define on [0,1]𝑘 ,𝑘 ∈ 𝑁, using the multilayered neural networks with radial basis functions. In [6] Li introduced a result for the simultaneous approximation using neural networks with radial basis functions, for a function and all it's derivatives. In[7] we can find a numerical approach for the results in [6].            The papers above used various weights in neural networks with radial basis, which make approximation theorems difficult in applications for engineering and any other science field.    Now there is a natural question: Can we use a fixed weight in radial basis function neural network, in order to have an easy usage for general usage in life?    Here let us answer the above question, and approximate any function in 𝐿𝑝spaces for 𝑝 < 1, using neural networks with radial basis with constant weight.We also construct an approximation for any function belongs to  𝐿𝑝 spaces for 𝑝 < 1 defined on a closed and bounded interval using neural networks with radial basis with constant weight. We introduce a direct estimation in 𝐿𝑝 spaces for 𝑃 < 1, and we define the ARTICLE INFO Submission date: 20/ 5 /2016 Acceptance date: 19 /9/ 2016 Publication date: 1/ 7/ 2019 Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019  401  convolution of two functions and prove a version of direct theorem by using it .We introduce a direct theorem for the approximation using radial basis function defined on bounded closed interval subset of the real line R.    There are various contributions to knowledge for the neural networks approximation, such as geophysics, metrology and graph rendering. In those applications, the data are all over sphere, and we need to represent them by a functional model.     2. Main Results 2-1 A Jackson Type Theorem We can approximate any function in  𝐿𝑝 , 𝑝 < 1 using radial basis function neural networks. If we want to make that possible we need some limits on the functions in 𝐿𝑝, 𝑝 < 1. As we see in our first result. Theorem 1. If 𝑓 ∈ 𝐿𝑝 , 𝑝 < 1 with lim|𝑥|→∞|𝑓(𝑥)| → 0 and 𝜎 be a radial basis function. Then there are real  constants ci, θ𝑖 and positive integers 𝑊 𝑎𝑛𝑑 𝑀 with ‖𝑓 − 𝑁‖𝑝 ≤  𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝, where 𝑁(𝑥) =∑𝑐𝑖𝜎(𝑊𝑥 + 𝜃𝑖).𝑀𝑖=1 Proof. By our hypotheses lim|𝑥|→∞|𝑓(𝑥)| = 0, ℎ > 0, ∃𝑘 ∈ 𝑍+such that  ‖∆ℎ𝑚𝑓(𝑥)‖𝐿𝑝(𝐽) ≤ 𝑐(𝑝)𝜔𝑚(𝑓, 𝛿)𝑝(𝐽)         𝑤ℎ𝑒𝑟𝑒|ℎ| < 𝛿  𝑤ℎ𝑒𝑛 𝐽 = (𝑘,∞) ∪ (−∞,−𝑘) ‖∆ℎ𝑚𝑓‖𝐿𝑝(𝐽) ≤ 𝑐(𝑝)𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐽)  Let            𝑀 = [1𝜔𝑚(𝑓,𝛿)𝑝] > 𝑘  𝑎𝑛𝑑 𝑀 > [1𝛿], Where [𝑦]𝑖𝑠(𝑡ℎ𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟 𝑝𝑎𝑟𝑡 𝑜𝑓 𝑦) + 1. Let 𝐼 =[−𝑀,𝑀]. If we devided I in to 2𝑀2 equidistance points , the distance between any two  points equal to 1 M⁄    𝐼1=[𝑥𝑜 , 𝑥1], 𝐼2 = [𝑥1, 𝑥2],⋯ 𝐼2𝑀2 = [𝑥2𝑀2−1, 𝑥2𝑀2] Let                                          𝜃𝑖 =𝑥𝑖+𝑥𝑖+12 , 𝑖 = 0,1,2,⋯ ,2𝑀2 − 1 Since 𝜎 is radial basis function, we can find 𝐿 > 0, such that ‖𝜎‖𝐿𝑝(𝐽) ≤𝑐(𝑝)𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐽) For 𝐼 = (𝐿,∞) ∪ (−∞, 𝐿).Define 𝑊 ∈ 𝑍+, 𝑤𝑖𝑡ℎ 𝑊 𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐼)⁄ > 𝐿. So we can define the following neural network  Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019  402  𝑁(𝑥) = ∑ ∆ℎ𝑚𝑓(𝑥𝑖)2𝑀2−1𝑖=0− 𝜎(𝑊(𝑥 − 𝜃𝑖)).    Then on 𝐶 = (−∞,−𝑀) ∪ (𝑀,∞)       ‖𝑓 − 𝑁‖𝐿𝑝(𝐶) = ‖𝑓‖𝐿𝑝(𝐶) + ‖𝑁‖𝐿𝑝(𝐶)                           ≤ 𝑐(𝑝)𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐶) + 𝑐(𝑝) ∑ ‖∆ℎ𝑚𝑓(𝑥𝑖)‖𝐿𝑝(𝐶) + ‖𝜎‖𝐿𝑝(𝐶)2𝑀2−1𝑖=0                                ≤ 𝑐(𝑝)𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐶) From now on 𝑐(𝑝) means a positive constant depends on 𝑝  only and may vary from one step to another.         Then on 𝐶 = [−𝑀,𝑀], there is 𝑗 = 0,1,2,⋯ ,2𝑀2 − 1 ,  with 𝐶 = ⋃ 𝐼𝑖2𝑀2−1𝑖=1  ‖𝑓 − 𝑁‖𝐿𝑝(𝐶) = ‖𝑓 − 𝑁‖𝐿𝑝(⋃ 𝐼𝑖2𝑀2−1𝑖=1 )                           ≤ 𝑐(𝑝) ‖𝑓 −∑∆ℎ𝑚𝑓(𝑥𝑖)𝜎(𝑊(𝑥 − 𝜃𝑖))𝑗−1𝑖=0‖𝐿𝑝(⋃ 𝐼𝑖2𝑀2−1𝑖=1 )                              + ‖ ∑ ∆ℎ𝑚𝑓(𝑥𝑖)𝜎(𝑊(𝑥 − 𝜃𝑖))2𝑀2−1𝑖=𝑗+1‖𝐿𝑝(⋃ 𝐼𝑖2𝑀2−1𝑖=1 )                          ≤ 𝐶(𝑝) 𝜔𝑚(𝑓, 𝛿)𝐿𝑝(𝐶). This completes the proof.  2-2 Convolution and its Approximation Let us recall the definition of the convolution of two functions 𝑓1, 𝑓2.      (𝑓1 ∗ 𝑓2)(𝑧) = ∫ 𝑓1(𝑦)𝑓2(𝑧 − 𝑦)𝑑𝑦𝑅 . Where 𝑧 is a real. Also consider the map 𝐹𝑘(𝑥) = {𝑑∆ℎ𝑘(𝑓(𝑥))𝑒− 11−𝑥2     |𝑥| < 1     0                            |𝑥| ≥ 1  𝑑 𝑖𝑠 a constant that make ∫ 𝐹𝑘𝑅  equal to one. It is clear 𝐹𝑘(𝑥) ∈ 𝐿𝑃0 (𝑅) ,then we use the definition of convolution to introduce a version of direct theorem.  Now let us define the space 𝐿𝑃0 (𝑅) = 𝐿𝑃 (𝑅) ∩ {𝑓: 𝑅 → 𝑅 𝑠𝑖𝑡 lim|𝑥|→0|𝑓(𝑥)| = 0} , we have the following result.  Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019  403  Theorem2. 𝐼𝑓 𝑓 ∈  𝐿𝑃0 (𝑅), then  ‖((𝐹𝐾 ∗ 𝑓) − 𝑓‖𝑝 ≤  𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝    Proof.               ‖((𝐹𝐾 ∗ 𝑓) − 𝑓‖𝑝 = ‖∫ 𝐹𝐾(𝑦)𝑓(𝑥 − 𝑦)𝑑𝑦 − ∫ 𝐹𝐾(𝑦)𝑓(𝑥)𝑑𝑦𝑅𝑅‖𝑝                                             = ‖∫ 𝐹𝐾(𝑦)𝑓(𝑥 − 𝑦)𝑑𝑦 + ∫ 𝐹𝐾(𝑦)𝑓(𝑥 − 𝑦)𝑑𝑦 − ∫ 𝐹𝐾(𝑦)𝑓(𝑥)𝑑𝑦𝑅|𝑦|>1|𝑦|<1‖𝑝                           ≤ (∫ (∫ |∆ℎ𝑘𝑓(𝑦)𝑒− 11−𝑦2𝑓(𝑥 − 𝑦)| 𝑑𝑦 + ∫ |𝐹𝐾(𝑦)𝑓(𝑥 − 𝑦)|𝑑𝑦|𝑦|>1 +             |𝑦|<1𝑅  ∫|𝐹𝐾(𝑦)||𝑓(𝑥)|𝑑𝑦)𝑝𝑅𝑑𝑥                          ≤   𝑐(𝑝) (∫ (∫ |∆ℎ𝑘𝑓(𝑦)|𝑑𝑦 + 0 + (∫ (∫ |∆ℎ𝑘𝑓(𝑦)|)𝑑𝑦)|𝑦|<1|𝑦|<1|𝑦|<1|𝑦|<1)𝑝𝑑𝑥                                             ≤ 𝑐(𝑝) (∫ |∆ℎ𝑘𝑓(𝑦)|𝑝𝑅𝑑𝑥)1 𝑝⁄                            ≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝 The proof is complete .   2-3 Approximation By Using Radial Basis Function In 𝑳𝒑 Spaces For  𝒑 <𝟏 With A Fixed Weight In this section we introduce a result by using radial basis function neural networks with fixed weight. as we see in the following theorem .  Theorem3. Let 𝑓 ∈ 𝐿𝑃0 ([𝑎1, 𝑎2] ), then If  𝜎 is a measurable radial basis function on R then  we can find real constants ciand θi and psitive integer 𝑊 𝑎𝑛𝑑 𝑀 with ‖𝑓 − 𝑁‖𝑝 ≤  𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝 Proof. Consider the real function: 𝑓(𝑥) ={            𝑓(𝑎1)𝑥 + (𝑎1 − 1)𝑓(𝑎1)                    𝑖𝑓𝑥 ∈ [𝑎1 − 1, 𝑎1]   𝑓(𝑥),                                                       𝑖𝑓 𝑥 ∈ [𝑎1, 𝑎2]−𝑓(𝑎2)𝑥 + (𝑎2 + 1)𝑓(𝑎2) ,              𝑖𝑓 𝑥 ∈ [𝑎2, 𝑎2 + 1]                  0                                       𝑖𝑓 𝑥 ∈ (−∞, 𝑎1 − 1] ∪ [𝑎2 + 1,∞) Using theorem 2 to get  ‖𝐹𝑘 ∗ 𝑓 − 𝑓‖𝑝 ≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝 ,so     ‖𝐹𝑘 ∗ 𝑓 − 𝑓‖𝑝 ≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝  on [𝑎1, 𝑎2]                     ( 2 − 1)  Since  ∫ 𝐹𝑘(𝑥 − 𝑦)𝑓(𝑦)𝑅  𝑑𝑦  is finite for any 𝑘 in ℤ+. And there exists a Riemann sum for the approximation of any convolution. If k is a positive integer we can find 𝑀𝑘 in ℤ+. And 𝑧𝑖 , 𝐶𝑖  for 𝑖 = 1,2,⋯ ,𝑀𝑘   satisfy Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019  404  ‖(𝐹𝑘 ∗ 𝑓)(𝑥) −∑𝐶𝑖𝐹𝑘(𝑥 − 𝑧𝑖)𝑓(𝑧𝑖)𝑀𝑘𝑖=1‖𝑝≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝                            (2 − 2) Using theorem 1 there exist real constants 𝛼𝑗,𝑘, 𝛽𝑗,𝑘  and  𝐾 ∈ ℤ+ satisfy  ‖𝐹𝑘(𝑥 − 𝑧𝑖) −∑𝛽𝑗,𝑘𝜎(𝑘(𝑥 − 𝑧𝑖)) + 𝛼𝑗,𝑘𝑗,𝑘‖𝑝≤ 𝑐(𝑝)𝜔𝑘(𝐹𝑘, 𝛿)𝑝                        (2 − 3)   Using eq. 2-1 to choose k ∈ ℤ+ that satisfy ‖𝑓(𝑥) − (𝐹𝑘 ∗ 𝑓)(𝑥)‖𝑝 ≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝                                                                (2 − 4) From eq. (2-2), (2-3) and (2-4) we obtain  ‖𝑓(𝑥) −∑𝐶𝑖𝑓~(𝑧𝑖)𝑀𝑘𝑖=1∑𝛽𝑗,𝑘𝜎(𝑘(𝑥 − 𝑧𝑖) + 𝛼𝑗,𝑘)𝑗,𝑘‖𝑝 ≤ ‖𝑓(𝑥) − (𝐹𝑘 ∗ 𝑓)(𝑥)‖𝑝 + ‖(𝐹𝑘 ∗ 𝑓)(𝑥) −∑𝐶𝑖𝐹𝑘(𝑥 − 𝑧𝑖)𝑓~(𝑧𝑖)𝑀𝑘𝑖=1‖𝑝 +‖∑𝐶𝑖𝐹𝑘(𝑥 − 𝑧𝑖)𝑓~(𝑧𝑖) −∑𝐶𝑖𝑓~(𝑧𝑖)𝑀𝑘𝑖=1𝑀𝑘𝑖=1∑𝛽𝑗,𝑘𝜎(𝑘(𝑥 − 𝑧𝑖) + 𝛼𝑗,𝑘 )𝑗,𝑘‖𝑝   ≤ 𝑐(𝑝)𝜔𝑘(𝑓, 𝛿)𝑝. This completes the proof.  3. Conclusion   There are main two problems in the field of studying approximation using neural networks, these problems are density and complexity. In our work here we consider the density problem. We proved a direct theorem for any function in 𝐿𝑝 spaces for 𝑝 < 1, satisfying lim|𝑥|→∞|𝑓(𝑥)| → 0, using neural networks with radial basis function. What distinguishes our work is the ability of neural networks approximation in approximation of functions. These thoughts lead to thoughts in the studying complexity approximation problem using fixed weight neural networks, as a future work.            Journal of University of Babylon for Pure and Applied Sciences, Vol. (27), No. (3): 2019  405  Conflict of Interests.  There are non-conflicts of interest .  References  [1] Chen, T.P and, Chen, H, “Approximation capability to function of several variables, nonlinear functionals, and operators by radial basis function neural networks”. IEEE trans. Neural Networks, 6: 904-910 ,1995.  [2] Leonard, J.A., Kramer, M.A. and Ungar, L.H., “Using radial basis functions to approximate a function and its error bounds”.. IEEE trans. Neural Networks, 3:622-630, 1992.  [3] Mhaskar, H.N. and Micchelli, C.A, “Approximation by superstition of sigmoidal and radial basis functions”. Advances Applied Mathematics, 13:350-373, 1992. [4] Park, J., and Sandberg, I.W, “Universal approximation using radial-basis –function networks”.  Neural Comput, 3:246-257, 1991. [5] Binfan, L. and Jennie, S, “The best approximation to 𝐶2 functions and its error bounds using regular center Gaussian networks” .IEEE trans. Neural Networks, 5: 845-847, 1994. [6] Li,X, “On simultaneous approximation by radial basis function neural networks”. Applied mathematics and computation, 95:75-89, 1998. [7] Duy, N.M. and Cong, T.T, “Approximation of function and its derivatives using radial basis function networks”. Applied Mathematics Modelling, 27:197-220, 2003.  اةصلاخل   ةيبصعلا تاكبشلا انمدختسا ثحبلا اذه يف .ةيبصعلا تاكبشلا مادختساب ةرمتسملا لاودلا بيرقت ثوحبلا نم ديدعلا تلوانت ءاضفلا يف لاودلا بيرقتلLp امدنع 𝑝 < 1بيرقت ةساردب انمق ذا .  ةيبصعلا تاكبشلا مادختساب لماكلا يقيقحلا طخلا ىلع ةفرعملا لاودلا.تباث نزو اهل يتلا انمدختسا امك  لاودلا بيرقتل فافتللاا ءاضفلا يفLp امدنع 𝑝 < 1  تاكبشلا مادختساب ةصوصرم ةرتف ىلع ةفرعم   .ةتباث نازوا اهل يتلا ةيبصعلا      ا تاملكلالدالة:  ءاضفلا𝐿𝑝  .فافتللاا ،ةيبصعلا تاكبشلا ،ديقملا بيرقتلا ،  

Approximation of Functions in L_P Spaces for p

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Approximation of Functions in L_P Spaces for p

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