<p>In this work, we develop the fixed-time TGNN (tensor gradient-based neural network) model for tackling the Sylvester tensor problem <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\sum _{n=1}^{N}\mathcal {X}(t)\times _{n}A_n=\mathcal {B}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mo>∑</mo> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>N</mi> </msubsup> <mi mathvariant="script">X</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>×</mo> <mi>n</mi> </msub> <msub> <mi>A</mi> <mi>n</mi> </msub> <mo>=</mo> <mi mathvariant="script">B</mi> </mrow> </math></EquationSource> </InlineEquation> online by generalizing the gradient-based design approach. We introduce a novel activation function for the TGNN model and theoretically prove its convergence. To demonstrate the superiority of our activation function, we compare it with four existing nonlinear activation functions within the TGNN model framework and provide convergence time upper bounds for each. Additionally, we validate our findings with three numerical examples.</p>