Development of Mamadu–Njoseh Polynomial Neural Networks for Physics-Informed Learning

2.1. Weighted space and orthogonality

Let the weight inner product and associated norm be defined as^[16]

where $w\left(x\right)$ is a prescribed weight function.

The MNP ${\left({\phi }_n\right)}_{n\ge 0}$ form a sequence of orthogonal polynomials such that

satisfying

and the normalisation condition

Using properties (2.2)–(2.4) with the weight $w\left(x\right)=1+x^2,$ the first three MNP are obtained as

${\phi }_0\left(x\right)=1,{\phi }_1\left(x\right)=x,{\phi }_2\left(x\right)=\frac{1}{3}\left(5x^2-2\right).$

Similarly,

${\left({\phi }_0\right)}_{L_w^2}^2=\frac{8}{3},{\left({\phi }_1\right)}_{L_w^2}^2=\frac{16}{15},{\left({\phi }_2\right)}_{L_w^2}^2=\frac{136}{189},$

which were obtained by exact polynomial integration on $\left(-1,1\right).$

Analogue to classical weighted Sobolev spaces, for integer $s\ge 0,$ define

$H_w^s\left(\left(-1,1\right)\right)=\left(u\in L_w^2:u^k\in L_{u_k}^2\text{for}0\le k\le s\right),$

where $w_k\left(x\right)$ denotes an appropriate family of derivative weights.^[17] The corresponding norm is

Because $\left({\phi }_n\right)$ are orthogonal in $L_w^2\left(\left(-1,1\right)\right),$ every $u\in L_w^2\left(\left(-1,1\right)\right)$ has the expansion

$u\left(x\right)=\sum _{n=0}^{\infty }{c}_n{\phi }_n\left(x\right),$

where the spectral coefficients are given by

$c_n=\frac{{\langle u,{\phi }_n\rangle }_w}{{\left({\phi }_n\right)}_{L_w^2}^2}.$

To facilitate analysis, we introduce the spectral Sobolev norm^[18]

where ${\left({\lambda }_n\right)}_{n\ge 0}$ is a sequence of positive numbers associated with the spectral scaling (typically related to eigenvalues of a Sturm–Liouville operator).

Let $S_N$ be the weighted orthogonal projection onto span $\left({\phi }_0,\dots ,{\phi }_N\right)$, that is,

If $u\in H_w^s$ with integer $s\ge 0,$ then for $0\le r\le s,$

${\left(u-S_{N}u\right)}_{H_w^r}\le k{N}^{r-s}{\left(u\right)}_{H_w^s},$

where $k>0$ is a constant. This result establishes algebraic convergence of the projection error for functions of Sobolev regularity.

Similarly, if $u$ is analytic in an ellipse of the complex plane that contains the interval $[-1,1],$ then the projection $S_{N}u$ achieves spectral convergence,^[19]

${\left(u-S_{N}u\right)}_{L_w^2}\le k{e}^{-\alpha N},$

for some constant $\alpha >0$ independent of $N.$

2.2. Mamadu-Njoseh polynomial neuron and layer

Let $H\subset L_w^2\left(\left(-1,1\right)\right)$ denote the weighted Hilbert space defined by

equipped with the weighted inner product

The MNP ${\left({\phi }_n\left(x\right)\right)}_{n=0}^{\infty }$ form a complete orthogonal basis for $H,\text{satisfying}$

where ${\delta }_{mn}$ is the Kronecker delta, and ${\gamma }_n$ is the normalisation constant associated with the nth basis function ${\phi }_n.$

A neuron in a MNP layer computes

where $W_j\in {ℝ}^{1\times d}$ and $b_j\in ℝ$ are trainable parameters for neuron $j$, and ${\phi }_n$ is the $nth$ MNP basis function.

Thus, the hidden transformation

$\Phi :{ℝ}^d\to {ℝ}^m$

is given^[20] by

which maps the input vector $x$ into a nonlinear feature space spanned by the orthogonal polynomial basis $\left({\phi }_0,\dots ,{\phi }_n\right)$,.

The operator $\Phi$ therefore enriches the representational capability of the network by embedding the input in a structured polynomial function space, ensuring controlled smoothness and improving numerical stability relative to generic activation functions.

2.3. MNP operator and polynomial neuron mapping

Each MNP ${\phi }_n\left(x\right)$ admits a finite polynomial expansion

where the coefficients $a_{n,k}$ are determined by the weighted orthogonality condition

${\int }_{-1}^1\left(1+x^2\right){\phi }_n\left(x\right){\phi }_m\left(x\right)dx=0,\text{for}\text{all}m\ne n.$

This ensures that ${\phi }_n\left(x\right)$ lies in the orthogonal complement of the polynomial subspace spanned by $\left({\phi }_0,\dots ,{\phi }_{n-1}\right)$.

Consequently, ${\phi }_n\left(x\right)$ can be constructed using properties (2.2)–(2.4) with the weight $w\left(x\right)=1+x^2$. Thus, each MNP basis element is generated by removing lower-order components, ensuring strict orthogonality in the weighted Hilbert space.

In the context of neural networks, each neuron of the form

$z_j=W_{j}x+b_j\Rightarrow h_j\text{}={\phi }_n\text{}\left(z_j\text{}\right),$

nonlinearly lifts the input into a higher-order orthogonal polynomial feature space. This performs an implicit orthogonal projection onto the $n-th$ spectral mode, enriching the representation and enhancing numerical stability relative to generic activations.^[20,21]

Define the MNP operator $\rho$ acting on an affine input

$z=Wx+b$

as

Thus, a single MNP neuron computes the nonlinear mapping

where $W_j$ and $b_j$​ are the weight and bias for neuron $j$.

Applying the binomial expansion yields

This representation shows that each MNP neuron induces a structured polynomial mapping of degree $n$ in the input variables. In particular, the terms ${\left(W_{j}x\right)}^r$ contain mixed products of input features, meaning that nonlinear cross-feature interactions are introduced through the multinomial expansion of $W_{j}x$.

2.4. Differential properties and learning dynamics of an MNP layer

Consider a neural network (or a single neuron) employing the $nth$ order MNP ${\phi }_n$​ as its activation function. For neuron $j$, define the affine transformation

$z_j\left(x\right)=W_{j}x+b_j,$

and the activation output

$h_j\left(x\right)={\phi }_n\left(z_j\left(x\right)\right),$

where the polynomial form of ${\phi }_n$​ is given by

Assuming the inputs $x\in {ℝ}^d$, the network output can be represented as

where the collection of trainable parameters is

$\Theta ={\left({\theta }_j,w_j,b_j\right)}_{j=1}^m,$

and ${\theta }_j\in ℝ$ are output-layer weights. The model thus forms a polynomial neural network where each neuron expands the input into an $n$-degree orthogonal polynomial basis component.

Since the pre‐activation for neuron $j$ is

$z_j=w_{j}x+b_j,$

the first and second derivatives of the MNP activation follow directly from the polynomial definition,

These closed forms enable efficient Jacobian and Hessian computation, critical for physics-informed neural network (PINN) training where higher–order derivatives appear in the PDE residual.

Parameter gradients for neutron $j$ can be computed as,

$\frac{\partial h_j}{\partial W_j}={\phi }_n^{\text{’}}\left(z_j\right)x^T\in {ℝ}^{1\times d},$

$\frac{\partial h_j}{\partial b_j}={\phi }_n^{\text{’}}\left(z_j\right)\in \mathrm{ℝ,}$

$\frac{\partial h_j}{\partial {\theta }_j}={\phi }_n\left(z_j\right)\in ℝ.$

Similarly, the gradient of the network output is given as,

$\begin{array}{c}{\nabla }_{u_j}F\left(x\right)={\theta }_j{\phi }_n^{\text{’}}\left(z_j\right)x^T\in {ℝ}^{1\times d},\\ \partial b_{j}F={\theta }_j{\phi }_n^{\text{’}}\left(z_j\right)\in ℝ,{\partial }_{\theta j}F={\phi }_n\left(z_j\right)\in ℝ.\end{array}$

2.5. Gradient magnitudes

Let $R>0$ be a bound on $\left(z_j\right)$ over the data distribution or during training. Define, for integer $0\le p\le n0$,

then the $pth$ derivative of ${\phi }_n$​ satisfies the uniform bound

$\underset{\left(\text{u}\right)\le \text{R}}{\max }\left({\phi }_n^{\left(p\right)}\left(u\right)\right)\le A_n^{\left(p\right)}\left(R\right)p!.$

The first–derivative bound of activation is defined by

This implies that if ${\left(x\right)}_2\le X,$ then the operator norm of the Jacobian w.r.t $W_j$ satisfies

Thus, the local Lipschitz constant of neuron $j$ with respect to $x$ is bounded by $\left(\left({\theta }_j\right)B_n\left(R\right)\right)$, where $B_n\left(R\right)$ controls derivative growth. Large $B_n\left(R\right)$ induces large gradients and potential instability, while small $B_n\left(R\right)$ may lead to vanishing gradients.

There are two primary mechanisms responsible for vanishing and exploding gradients in MNP–based networks:^[1]

• a.

  Since ${\phi }_n^{\text{'}}\left(u\right)$ is a polynomial of degree ($n-1)$, the following scenario exists;

• i.

  If $\left(z\right)\ge 1$ and coefficients are moderate, ${\phi }_n^{\text{'}}\left(z\right)$ can grow as $O\left({\left(z\right)}^{n-1}\right)$, implying exploding gradients.

• ii.

  If $\left(z\right)\le 1$ and higher–order coefficient dominate but small, ${\phi }_n^{\text{'}}\left(z\right)\approx a_{n,1}$ implying small gradients.

• b.

  Gradients for $W_j$ scale with $\left({\theta }_j\right)$ implies the following possibilities;

• i.

  Large $\left({\theta }_j\right)$ amplify, and

• ii.

  Regularising or initialising ${\theta }_j$ small mitigates an explosion.

A practical bound of the gradient to avoid explosion is to enforce^[22]

such that

$\left({{\phi }^{\prime }}_n\left(z\right)\right)\le Gforall\left(z\right)\le R.$

However, (3.18) can be satisfied by a scaling coefficient $a_{n,k}$ or enforcing $R$ small via weight normalisation.

2.6. MNP-PINN construction

Let $\Omega \subset {ℝ}^d$ be the spatial domain and $I=\left(0,T\right)$ the time interval. Consider a partial differential equation (PDE) operator $\aleph$ acting on the scalar field $u\left(x,t\right)$ defined as

with boundary conditions $B\left(u\right)=g$ on $\partial \text{Ω}\times I$ and initial conditions $u\left(x,0\right)=u_0\left(x\right),{\partial }_{t}u\left(x,0\right)=u_1\left(x\right).$ $f\left(x,t\right)$ is the source term.

We seek an approximate $u_{\theta }$ parametrized by $\theta$ that satisfies the PDE and boundary/initial conditions, in an informed sense.

Since ${\left({\phi }_n\left(\xi \right)\right)}_{n=0}^{\infty }$ is orthogonal on $\left(-1,1\right)$ w.r.t. $w\left(\xi \right)=1+{\xi }^2$ and ${\phi }_n\left(1\right)=1$, then for any pre-activation $\xi =\varnothing \left(x,t;{\theta }_{\varnothing }\right)$ we define a polynomial layer

${\wp }_N\left(\xi \right)=\sum _{n=0}^N{\alpha }_n{\phi }_n\left(\xi \right),$

with learned coefficients ${\alpha }_n={\alpha }_n\left(\theta \right).$

The choice of network representation determines how the neural architecture captures and expresses the spatial-temporal dependencies in the solution $u_{\theta }\left(x,t\right)$. Two formulations are commonly considered.

• a.

  Spectral form: The spectral form defines a global functional expansion of the neural approximation $u_{\theta }\left(x,t\right)$ over a set of basis functions ${\left({\phi }_n\left(x,t\right)\right)}_{n=0}^N$​. It is expressed as

where $c_n$ are small neural networks that modulate each function.

• b.

  Activation form: Alternatively, the activation form incorporates the orthogonal polynomial basis directly as an activation function within the hidden layers of the network. It is defined as

such that the orthogonal polynomial structure is embedded in the layer.

In this paper, for analytical clarity and interpretability, the spectral form (4.2) is adopted. Under this formulation, the approximation is rewritten as:

where $\xi$ is an affine map $\xi =B_{x}x+B_{t}t+b,$ or normalised coordinate mapping $\xi =M\left(x,t\right)\in \left(-1,1\right).$ $B_x,B_t,b$ are Learnable scaling and shift parameters used to normalise spatial and temporal coordinates into the standard domain [−1,1] [Figure 1].

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Figure 1:

Architecture of the proposed Mamadu-Njoseh polynomial physics informed neural network (MNP-PINN).

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The MNP-PINN is implemented as a fully connected feedforward neural network. The network consists of one hidden layer with 40 neurons. The hidden layer employs quadratic Mamadu–Njoseh polynomials (MNPs) as activation functions, providing orthogonality and stability benefits over conventional activations. The output layer is linear to produce predicted solution values, as shown in Figure 1. The input dimension is $d=1$ for the one-dimensional Poisson problem.

2.7. Physics-informed loss

Let ${\left(\left(x_i,t_i\right)\right)}_{i=0}^{N_r}\subset \Omega \times I$ be sampled collocation points for PDE residuals, boundary points $\left(\left(x_j,t_j\right)\right)\subset \partial \Omega \times I$, and initial points ${\left(x_k\right)}_{k=1}^{N_0}\subset \Omega$. The physics-informed loss is given as

where,

$L_{res}=\frac{1}{N_r}\sum _{i=1}^{N_r}{\left(M\left(u_{\theta }\right)\left(x_i,t_i\right)-f\left(x_i,t_i\right)\right)}^2,$

$L_{bnd}=\frac{1}{N_b}\sum _{j=1}^{N_b}{\left(B\left(u_{\theta }\right)\left(x_j,t_j\right)-g\left(x_j,t_j\right)\right)}^2,$

$L_{init}=\frac{1}{N_0}\sum _{k=1}^{N_0}{\left(u_{\theta }\left(x_k,0\right)-u_{\theta }\left(x_k\right)\right)}^2.$

Here, $L_{res}$, $L_{bnd}$, and $L_{init}$ denotes PDE residual loss, boundary condition loss, and initial condition loss, respectively, to ensure consistency, stabilization and smoothness. $N_r$, $N_b$, $N_0$ denotes the total number of residual points, boundary points, and initial points, respectively. The weighting parameters ${\lambda }_b,{\lambda }_0,{\lambda }_{reg}>0$, are positive scaling coefficients that control the relative importance of boundary, initial, and regularization losses, respectively.

Since the basis functions ${\phi }_n\left(\xi \left(x,t\right)\right)$ are polynomials of the transformed coordinate $\xi \left(x,t\right)$, their derivatives with respect to $t$ and $x$ can be computed analytically through the chain rule. Hence,

If $c_n$ depends on $\left(x,t\right)$, the polynomial structure reduces numerical error in derivative evaluation relative to generic activation because ${{\phi }^{\prime }}_n\left(\xi \right)$ are known polynomials of degree $\left(n-1\right)!.$ When the coefficient functions $c_n$ are allowed to depend on both spatial and temporal variables, i.e., $c_n=c_n\left(x,t;{\theta }_c\right)$, the spectral representation becomes semi-local, combining data adaptability with the analytical structure of the orthogonal basis.

2.8. Network initialisation

An MNP-PINN of depth $l$ consists of layers

where

$z^{\left(l\right)}=W^{\left(l\right)}{h}^{\left(l-1\right)}+b^{\left(l\right)},$

with $h^{\left(0\right)}=x\in {ℝ}^{d_0},W^{\left(l\right)}\in {ℝ}^{d_l\times d_{l-1,}}$ and $b^{\left(l\right)}{ℝ}^{d_l}$, where we choose distributions for $W^{\left(l\right)}$ and $b^{\left(l\right)}$ such that

• i.

  $h^{\left(l\right)}$ neither collapses to 0 nor blows up (non-degenerate forward pass).

• ii.

  Gradients $\partial L/\partial W^{\left(l\right)}$ remain well-scaled across layers.

• iii.

  The MNP orthogonality structure and its weighted normalisation are preserved (at least initially), so polynomial derivatives behave as expected.

Let’s analyse the propagation of the activation statistics through layers. Assume input $x_i$ have zero mean and variance ${\sigma }_x^2$, and weights are independent and identically distributed $\left(i.i.d.\right)$ random variables with

Then, for neutron $i$ in layer $l$, we have that

Assume $h_j^{\left(l-1\right)}$ are $i.i.d$. with $E\left(h_j^{\left(l-1\right)}\right)=0$ and $\text{Var}\left(h_j^{\left(l-1\right)}\right)=q_{l-1}$. Also let$,\text{Var}\left(b_i^{\left(l\right)}\right)={\sigma }_b^2$. Then by independence, we get

For a nonlinear activation, let the activation at layer $l$ be $h^{\left(l\right)}={\phi }_{nl}\left(z^{\left(l\right)}\right).$ Define the second moment map as

$M_{2,n}\left({\sigma }_z^2\right)=E_{Z\sim N\left(0,{\sigma }_z^2\right)}\left({\phi }_n^2\left(z\right)\right).$

Then the variance of post-activation is

For a symmetric input distribution, we can approximate using even-order coefficients of ${\phi }_n$, given as

where $a_{n,k}$ are the MNP coefficients.

If $z^{\left(l\right)}\sim N\left(0,{\sigma }_z^2\right),$ then

$E\left({\left(z^{\left(l\right)}\right)}^{2k}\right)=\left(2k-1\right)!!{\sigma }_z^{2k}.$

To prevent variance blow-up, we impose energy conservation between layers by setting

Substituting (5.7) into ((5.6) gives

$d_{l-1}{\sigma }_w^2\sum _{k=1}^n{a}_{n,k}^2\left(2k-1\right)!!{\sigma }_z^{2k-2}=1.$

Hence,

which defines the critical weight variance to ensure stability. To this effect, we set ${\sigma }_z^2={\sigma }_x^2$ for the first layer. Deeper layers can be recursively estimated.

For instance, let us consider a special case involving the second-order MNP ${\phi }_n\left(x\right)=\frac{1}{3}\left(5x^2-2\right)$. We have the coefficients $a_{2,2}=\frac{5}{3},a_{0,2}=-\frac{2}{3}$. Then,

$\text{Var}\left(h^{\left(l\right)}\right)\approx {\left(\frac{5}{3}\right)}^{2}E\left(z^4\right)=\frac{25}{3}{\sigma }_z^4.$

Hence, the critical initialisation becomes

${\sigma }_w^2=\frac{1}{d_{l-1}\frac{25}{3}{\sigma }_z^2}.$

Setting ${\sigma }_z^2=1$ arbitrarily, we have

${\sigma }_w^2=\frac{3}{25d_{l-1}},$

which ensures stable propagation for quadratic MNP activations. Also, quadratic activations magnify magnitudes more strongly, so weights must be initialised smaller to maintain stable variance.

3.1. Numerical example

We demonstrate the accuracy of the proposed method by considering a standard one-dimensional Poisson problem. Let $u:\left(-1,1\right)\to ℝ$ satisfy

$-u^{″}\left(x\right)={\pi }^2\sin \left(\pi x\right),x\in \left(-1,1\right),$

subject to the homogeneous Dirichlet boundary conditions

$u\left(-1\right)=u\left(1\right)=0.$

The analytical solution of this boundary value problem is

$u\left(x\right)=\sin \left(\pi x\right).$

This example is smooth and well-posed, and therefore serves as a suitable benchmark for testing both accuracy and convergence behaviour of the proposed Mamadu–Njoseh-based Physics-Informed Neural Network (MNP-PINN) framework. The aim is to approximate $u\left(x\right)$ numerically over $\left(-1,1\right)$ and compare the obtained solution with the exact one in terms of the $L_2$​ and $L_{\infty }$ error norms. The coefficients $c_n,n=0,\dots ,N$, the lifted-basis multiplier $l\left(x\right)$, and the training parameters are chosen to effectively control the dynamics of the equation.

All network parameters are initialised according to Table 1 to ensure stable and efficient training:

Based on the initialisation and configuration parameters summarized in Table 1 for the MNP-PINN framework, the ensuing results are obtained via Python implementation and are presented in the subsequent tables and figures.

The network is trained using the Adam optimiser with an initial learning rate of 1×${10}^{-6}$, decaying exponentially by a factor of 0.5 every 1000 epochs. The total training duration is 3000 epochs. Loss function and error convergence at selected epochs are reported in Table 2, demonstrating rapid convergence and high-accuracy approximations with $L_2$​ and $L_{\infty }$​ errors on the order of ${10}^{-5}$.

Collocation points are sampled uniformly within the computational domain $\left(-1,1\right)$, resulting in 200 interior points. Boundary points are fixed at the endpoints $x=-1$ and $x=1$, enforcing Dirichlet boundary conditions. This procedure provides sufficient coverage of the domain for the 1D Poisson problem while maintaining computational efficiency.

3.2. Discussion

Figure 2 depicts the performance of the MNP-PINN applied to the 1D Poison Problem, illustrating the model’s ability to approximate the analytical solution of the target problem with remarkable precision. The figure typically shows both the predicted solution obtained by the MNP-PINN and the exact analytical solution, often plotted together for direct visual comparison. The near-perfect overlap between predicted solution and exact solution across the entire spatial domain demonstrates the spectral accuracy of the MNP-PINN. This high degree of correspondence implies that the MNP activation functions successfully capture the underlying smoothness and structure of the solution, even in regions with steep gradients or boundary effects. Additionally, the residual error often visualised below the main plot remains close to zero throughout, confirming numerical stability and robust convergence during training. The network’s ability to generalise well on such benchmark examples underscores the effectiveness of MNP layers in achieving physics-consistent approximations for differential problems, outperforming traditional neural architectures that rely on standard nonlinear activations.

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Figure 2:

Mamadu-Njoseh polynomial physics informed neural network (MNP-PINN) for 1D poison problem. X represents the independent variable, u(x) represents the dependent variable. MNPNN: Mamadu–Njoseh polynomial neural network..

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Table 2 summarizes the training performance of the MNP-PINN across selected epochs for the 1D Poisson equation. The table records the evolution of the loss function, the L₂ norm error, and the L∞ norm error throughout the optimisation process. The results show a progressive and stable convergence pattern. Initially, the loss begins at a relatively high value (4.854×10¹) and decreases by nearly seven orders of magnitude, reaching values around 10 ⁻⁶ after sufficient training. This sharp decline signifies efficient error minimization and well-conditioned gradient flow within the MNP-PINN framework. Both the L₂​ and L∞​ error norms exhibit consistent reduction, converging to approximately 10⁻⁵, which corresponds to spectral-level accuracy, which is a hallmark of high-order polynomial approximation. A minor oscillation in the error curve around epoch 1000 reflects transient optimiser nonlinearity or adaptive learning rate adjustment, but the model rapidly stabilizes thereafter, reaffirming its robust training dynamics. Overall, the table confirms that the MNP-PINN achieves high numerical precision, strong generalisation, and stable convergence behaviour, validating the effectiveness of MNP activation structures in solving partial differential equations like the 1D Poisson problem.

Figure 3 presents the coefficient spectrum of the MNP-PINN, showing the magnitude decay of the polynomial coefficients $\left(c_n\right)$ as a function of the polynomial degree n, this spectrum provides insight into the representational efficiency and smoothness of the learned solution. As $n$ increases, the coefficients $\left(c_n\right)$ exhibit a rapid exponential-like decay, indicating that the MNP-PINN effectively captures the dominant low-order modes of the solution while suppressing higher-order components. This behaviour confirms spectral convergence, meaning that the error decreases faster than any algebraic rate as the polynomial order increases. Such rapid coefficient attenuation signifies that higher-degree Mamadu–Njoseh basis functions contribute negligibly to the final approximation, thereby ensuring a compact spectral representation. This property reduces overfitting and numerical oscillations, common issues in classical neural architectures using purely analytical activations. Consequently, the MNP-PINN achieves smooth, stable, and physically consistent approximations across the spatial domain, showcasing the advantage of embedding Mamadu–Njoseh orthogonal structures within the neural framework.

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Figure 3:

Coefficient spectrum of Mamadu-Njoseh polynomial physics informed neural network (MNP-PINN) for the 1D poison problem.

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Figure 4 presents the pointwise absolute error distribution between the MNP-PINN-predicted solution $\hat{u}\left(x\right)$ and the analytical solution u(x) across the spatial domain. The error is defined as

$E\left(x\right)=\left(\hat{u}\left(x\right)-u\left(x\right)\right),$

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Figure 4:

Pointwise absolute error of Mamadu-Njoseh polynomial physics informed neural network (MNP-PINN) for the 1D poison problem. x represents the independent variable.

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where $\hat{u}\left(x\right)$ is the computed solution, and $u\left(t\right)$ is the exact solution. The plot reveals that the MNP-PINN achieves uniformly low errors throughout the domain, with the maximum deviation confined to the boundary regions where solution gradients are typically steeper. The central region maintains near-zero error, highlighting the network’s strong approximation capacity and stability. The observed smoothness and near-symmetry of $E\left(x\right)$ reflect the polynomial orthogonality and spectral convergence characteristics of the Mamadu–Njoseh basis functions. Overall, the error magnitude remains on the order of 10⁻⁵, confirming high numerical accuracy and demonstrating the capability of MNP-PINN to resolve fine-scale spatial structures in the Poisson solution with minimal numerical dispersion.

Figure 5 illustrates the evolution of the MNP-PINN during training. The top panel shows the training loss decreasing exponentially by several orders of magnitude, confirming rapid and stable convergence. The lower four panels present MNP-PINN predictions at selected epochs (initial, early, mid, and final), revealing how the network progressively refines its approximation until achieving near-perfect overlap with the analytical solution. Minor oscillations in the loss reflect optimiser nonlinearity but do not affect overall stability. By the final epoch, both L₂​ and L∞​ errors reach the order of 10⁻⁵, demonstrating spectral-level accuracy and excellent generalisation. The figure highlights the MNP-PINN’s ability to efficiently learn smooth PDE solutions through polynomial-based activation structures.

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Figure 5:

Training progression of Mamadu-Njoseh polynomial physics informed neural network (MNP-PINN) for the ID poison problem. u(x) represents the dependent function. x represents the independent function.

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