<p>A Bernstein polynomial-based neural network framework is developed to approximate the solution of a biologically motivated time-fractional tumor-immune-chemotherapy model. The model consists of four coupled Caputo-type fractional diffusion–reaction equations describing the spatial–temporal dynamics of normal cells, tumor cells, immune cells, and chemotherapeutic drug concentration in a one-dimensional tissue domain, with parameters chosen according to established tumor–immune–drug interaction models. A tensor-product Bernstein feature map in space and time is used as input to a deep fully connected network, and the governing fractional partial differential equations are enforced through a composite loss that combines a vectorized L1 finite-difference approximation for the Caputo time derivatives with automatic differentiation for the spatial second derivatives. The network is trained in high precision using an Adam-LBFGS pipeline and is validated first on a manufactured solution, for which the forcing terms are generated consistently with the same discrete operators, yielding mean absolute errors below <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$10^{-4}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>4</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> and maximum absolute errors below <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$10^{-3}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mn>10</mn> <mrow> <mo>-</mo> <mn>3</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> on dense evaluation grids for several fractional orders <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\alpha \in (0,1)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Additional examples having explicit exact solution, benchmarked against a refined L1 finite-difference reference, demonstrate that the Bernstein neural network retains good accuracy and exhibits stable performance with respect to changes in the fractional order, confirming its robustness as a solver for time-fractional tumor-immune-chemotherapy systems.</p>

A bernstein polynomial-based neural network framework for solving fractional tumor growth mathematical model

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发表日期：

2026年3月2日

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Paras Varshney,
Govind Raj Naunyal,
Sachin Kumar

摘要

A Bernstein polynomial-based neural network framework is developed to approximate the solution of a biologically motivated time-fractional tumor-immune-chemotherapy model. The model consists of four coupled Caputo-type fractional diffusion–reaction equations describing the spatial–temporal dynamics of normal cells, tumor cells, immune cells, and chemotherapeutic drug concentration in a one-dimensional tissue domain, with parameters chosen according to established tumor–immune–drug interaction models. A tensor-product Bernstein feature map in space and time is used as input to a deep fully connected network, and the governing fractional partial differential equations are enforced through a composite loss that combines a vectorized L1 finite-difference approximation for the Caputo time derivatives with automatic differentiation for the spatial second derivatives. The network is trained in high precision using an Adam-LBFGS pipeline and is validated first on a manufactured solution, for which the forcing terms are generated consistently with the same discrete operators, yielding mean absolute errors below $10^{-4}$ $10^{- 4}$ and maximum absolute errors below $10^{-3}$ $10^{- 3}$ on dense evaluation grids for several fractional orders $\alpha \in (0,1)$ $α \in (0, 1)$ . Additional examples having explicit exact solution, benchmarked against a refined L1 finite-difference reference, demonstrate that the Bernstein neural network retains good accuracy and exhibits stable performance with respect to changes in the fractional order, confirming its robustness as a solver for time-fractional tumor-immune-chemotherapy systems.