<p>Non-linear convection–reaction–diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of these non-linear systems is essential due to the challenges of obtaining exact solutions. Traditionally, the Galerkin finite element method (GFEM) has been the standard computational tool for solving these PDEs. With the advancements in machine learning, Physics-informed neural network (PINN) has emerged as a promising alternative for approximating non-linear PDEs. In this study, we compare the performance of the PINN and GFEM by solving four one-dimensional non-linear CRD problems with varying initial and boundary conditions. We evaluated PINN’s performance relative to GFEM using key metrics, including the absolute error <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Vert \Delta \Vert _1 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, maximum absolute error <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$ l_{\infty } $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ l_2 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-norm, along with visual representations and statistical methods such as the root mean squared error (RMSE), the standard deviation, the Wilcoxon signed-rank test (WSRT), and the coefficient of variation (CV). Our findings reveal that while both methods achieve solutions close to the analytical results, PINN demonstrated superior accuracy and efficiency, achieving significantly lower <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">$ \Vert \Delta \Vert _1 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mrow> <mo stretchy="false">‖</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">‖</mo> </mrow> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq5.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$l_{\infty }$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>∞</mi> </msub> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40819_2025_1904_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">$ l_2 $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>-norm values. Statistical analysis shows that PINN attained smaller RMSE and reduced standard deviations for the Burgers’ equation, Fisher’s equation, and Newell–Whitehead–Segel equation, indicating higher accuracy and consistency. Although GFEM shows slightly better accuracy for the Burgers–Huxley equation, it is less consistent over time. In contrast, PINN exhibits more reliable and robust performance, underscoring its potential as a cutting-edge approach for solving nonlinear PDEs.</p>

From Mesh to Neural Nets: A Multi-method Evaluation of Physics Informed Neural Network and Galerkin Finite Element Method for Solving Nonlinear Convection–Reaction–Diffusion Equations

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2025年4月30日

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Fardous Hasan,
Hazrat Ali,
Hasan Asyari Arief

摘要

Non-linear convection–reaction–diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of these non-linear systems is essential due to the challenges of obtaining exact solutions. Traditionally, the Galerkin finite element method (GFEM) has been the standard computational tool for solving these PDEs. With the advancements in machine learning, Physics-informed neural network (PINN) has emerged as a promising alternative for approximating non-linear PDEs. In this study, we compare the performance of the PINN and GFEM by solving four one-dimensional non-linear CRD problems with varying initial and boundary conditions. We evaluated PINN’s performance relative to GFEM using key metrics, including the absolute error $ \Vert \Delta \Vert _1 $ ${‖ Δ ‖}_{1}$ , maximum absolute error $ l_{\infty } $ $l_{\infty}$ , and $ l_2 $ $l_{2}$ -norm, along with visual representations and statistical methods such as the root mean squared error (RMSE), the standard deviation, the Wilcoxon signed-rank test (WSRT), and the coefficient of variation (CV). Our findings reveal that while both methods achieve solutions close to the analytical results, PINN demonstrated superior accuracy and efficiency, achieving significantly lower $ \Vert \Delta \Vert _1 $ ${‖ Δ ‖}_{1}$ , $l_{\infty }$ $l_{\infty}$ , and $ l_2 $ $l_{2}$ -norm values. Statistical analysis shows that PINN attained smaller RMSE and reduced standard deviations for the Burgers’ equation, Fisher’s equation, and Newell–Whitehead–Segel equation, indicating higher accuracy and consistency. Although GFEM shows slightly better accuracy for the Burgers–Huxley equation, it is less consistent over time. In contrast, PINN exhibits more reliable and robust performance, underscoring its potential as a cutting-edge approach for solving nonlinear PDEs.