[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82266-en":3,"doc-seo-82266-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82266,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","All you need is SAMPAT","Deep neural networks underpin much of current AI/ML, yet they typically lack interpretability, limiting scientific insight when analyzing experimental data. SAMPAT (Smooth Approximation via Multivariate Polynomials and Analytic Transformations) introduces a three-layer neural architecture provably capable of learning continuous, everywhere differentiable functions and approximating smooth functions arbitrarily closely. Its output admits a closed, compact algebraic analytic form, enabling complete interpretability. Experiments on synthetic and benchmark datasets show competitive results with simpler representations, and a two-layer variant often suffices.","All you need is SAMPAT  \nJayadeva and Madhur Aswani  \nDepartment of Electrical Engineering, Indian Institute of Technology, Delhi  \narXiv :2607 .09235v 1 [ cs .LG] 10 Jul 2026  \nAbstract  \nThe current state of the art in AI/ML rests on deep neural architectures, which, in general, suffer from a lack of interpretability.  \nInterpretability is crucial to gleaning insights while analyzing experimental data, where quantitative predictions may not be adequate for a scientist. We present a three layer neural architecture, SAMPAT (Smooth Approximation via Multivariate Polynomialsand Analytic Transformations), that can provably learn a continuous, everywhere differentiable function, that can approximate any smooth function arbitrarily closely. SAMPAT’s approximant can be expressed as a closed and compact algebraic, analytic expression, providing complete interpretability. Experiments on synthetic and benchmark datasets indicate that SAMPAT yields competitive performance with simpler representations. For many tasks, a two layer SAMPAT suffices. By imposing restrictions on the connectivity between neurons, SAMPAT may be used to provide a range of approximants, including regular and trigonometric polynomials, rational expressions, Gaussians, mixtures of Gaussians, as well as arbitrary combinations of the same; without restrictions, it learns a suitable structure. SAMPAT may be used to factorize polynomials and model nonlinear systems. With the addition of skip connections, a 4 to 6 layer SAMPAT is adequate to represent a substantive range of methods widely used in AI/ML, allowing the choice of a model’s family, not just its parameters, to also be optimized as part of the learning process.  \nIndex Terms  \nNeural networks, AI, machine learning, interpretability, poynomials.  \nI. INTRODUCTION  \nIt is widely known that a three layer neural network can learn any smooth input-output map. Much work has been done on existence proofs [1] . Most of these relate to functions on the unit n-dimensional cube. However, constructing a three layer network for a given task has been elusive, and most trained networks in use are accurate, black box predictors. We propose SAMPAT (Smooth Approximation using Multi-Polynomial and Analytic Transformations), a 3 layer neural architecture, that facilitates the construction of interpretable models and their analysis. Figure 1 describes the basic SAMPAT architecture.  \nFig. 1: Basic SAMPAT architecture  \nInputs to layer 1 neurons are denoted by x 1 , x2 , ... xn or xi , i = 1 , 2, ..., n. The weight connecting input xj to the i-th first layer neuron is uij ; the notation assumes the destination i is the first letter of the subscript and j is the source. The net input to the i-th first layer neuron, net1i is given by  \nn  \nnet1i = X uij xj + bi (1)  \nj=1  \nwhere bi is a bias term that is added to the weighted sum. Without loss of generality, we assume that inputs include a constant input with a value of 1; a weight of bi associated with such an input serves the same purpose. Hence, unless required otherwise, we will drop the bias term to keep the notation and descriptions facile. First layer neurons have a logarithm activation function. The output of the i-th first layer neuron is denoted by yi , and is given by  \nn  \nyi = log (net1i) = log(X uij xj) (2)  \nj=1  \nThe base of the logarithm is normally e ≈ 2.71828, except for digital circuit implementations, where base 2 is desirable. Note that log2 (net1i) = logloeg(ne(e2t)1~~i~~) , and loge (net1i) = loglo2g(n2(eet)1~~i~~) , implying that changing the base is equivalent to a gain term. Layer2  \n0000–0000/00$00.00 © 2021 IEEE  \nneurons receive weighted sums of layer 1 outputs. The weight of the connection from neuron i in layer 1 to neuron k in the layer 2 is denoted by vki. The net input to neuron k in layer 2 is denoted by net2k and is given by  \nnet2k = AX vki yi = AX vki log 􀀰 nX uij xj 􀀱 = AX log 􀀰 nX uij xj 􀀱 vki = log 􀀰 AY 􀀰 nX uij xj 􀀱 vki 􀀱 (3)  \ni=1 i=1 􀁀j=1 ","cbCaimSblVPkgyXg","https://ap.wps.com/l/cbCaimSblVPkgyXg","pdf",1003730,1,7,"English","en",105,"# Abstract\n# Index Terms\n# Introduction\n## Three-layer SAMPAT overview\n## Layer 1: logarithm activation\n## Layer 2: exponential activation and reducible polynomials\n## Layer 3: linear combinations for irreducible polynomials","[{\"question\":\"What problem does SAMPAT address in current AI/ML models?\",\"answer\":\"SAMPAT targets the interpretability gap of deep neural architectures, aiming to provide analytic, inspectable models rather than black-box predictors.\"},{\"question\":\"What is the core representational idea behind SAMPAT?\",\"answer\":\"SAMPAT constructs approximants using a structured three-layer design: logarithm-based layer outputs, exponential activations yielding reducible polynomials, and a third layer that forms linear combinations to capture broader function families.\"},{\"question\":\"How does SAMPAT support interpretability and approximation guarantees?\",\"answer\":\"SAMPAT’s approximant can be written as a closed and compact algebraic analytic expression, and the architecture is described as provably able to learn continuous, everywhere differentiable functions with arbitrary closeness for smooth 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problem does SAMPAT address in current AI/ML models?","Question",{"text":74,"@type":75},"SAMPAT targets the interpretability gap of deep neural architectures, aiming to provide analytic, inspectable models rather than black-box predictors.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"What is the core representational idea behind SAMPAT?",{"text":79,"@type":75},"SAMPAT constructs approximants using a structured three-layer design: logarithm-based layer outputs, exponential activations yielding reducible polynomials, and a third layer that forms linear combinations to capture broader function families.",{"name":81,"@type":72,"acceptedAnswer":82},"How does SAMPAT support interpretability and approximation guarantees?",{"text":83,"@type":75},"SAMPAT’s approximant can be written as a closed and compact algebraic analytic expression, and the architecture is described as provably able to learn continuous, everywhere differentiable functions with arbitrary closeness for smooth 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