Memory-Informed Neural Pseudo-Operator (MINPO)

• MINPO is a mesh-free neural framework that models nonlocal spatiotemporal dynamics and fractional PDEs via learned memory operators and inverse reconstructions.
• It integrates MLP and Kolmogorov-Arnold networks as encoders with a nonlocal-consistency loss to enforce coherence between the learned operator and the reconstructed dynamics.
• Benchmark experiments show MINPO achieves significant error reductions and speedups compared to classical solvers and existing PINN approaches.

The Memory-Informed Neural Pseudo-Operator (MINPO) is a mesh-free neural framework designed to model and resolve nonlocal spatiotemporal dynamics described by integro-differential equations (IDEs). It encapsulates both the memory operator and its inverse through neural networks, enabling explicit reconstruction of unknown solution fields. The architecture leverages either multilayer perceptrons (MLPs) or Kolmogorov-Arnold Networks (KANs) as encoders and introduces a nonlocal consistency loss to enforce coherence between the learned operator field and the reconstructed dynamics. MINPO is applicable to systems governed by diverse nonlocal kernel structures, including fractional partial differential equations (PDEs), and demonstrates robust generalization across problem classes and kernel dimensionalities (Mostajeran et al., 19 Dec 2025).

1. Mathematical Formulation of Nonlocal Dynamics

MINPO targets the general form of an IDE embodying spatial nonlocality, temporal memory, and fractional dynamics. The governing equation in space-time coordinates $\xi=(x,t)\in\Omega\times(0,T)$ is:

$T_\alpha[u](\xi) = N[u](\xi) + M[u](\xi) + S(\xi)$

• $N[u](\xi)$: local (possibly nonlinear) differential operator
• $T_\alpha$: composite time-evolution operator, including the first-order and fractional Caputo derivative,

$$T_\alpha[u](\xi) = \lambda_1 \frac{\partial u}{\partial t}(\xi) + \lambda_\alpha \, {}^C D_t^\alpha u(\xi), \qquad 0<\alpha < 1$$

where ${}^C D_t^\alpha$ is the Caputo derivative,

$${}^C D_t^\alpha u(x,t) = \frac{1}{\Gamma(1-\alpha)} \int_0^t \frac{\partial u(x,\tau)}{\partial \tau} (t-\tau)^{-\alpha}~d\tau$$

• $M[u](\xi)$: unified nonlocal memory operator, for general kernel $K$,

$$M[u](\xi) := \int_{\Omega_{\xi^-}} K(\xi,\eta;u(\eta))\,d\eta$$

where $\Omega_{\xi^-}$ denotes the causal past of $\xi$.

The inverse of the Caputo derivative is the Riemann–Liouville fractional integral:

$$I_t^\alpha[f](t) = \frac{1}{\Gamma(\alpha)} \int_0^t (t-\tau)^{\alpha - 1} f(\tau)\,d\tau$$

Many classical and fractional nonlocal PDEs are subsumed as special cases of this general structure.

2. Neural Architecture and Operator Representation

MINPO learns the memory operator and its inverse via two neural networks:

• Memory encoder $M_\theta(\xi) \approx M[u](\xi)$
• Inverse-memory encoder $$J_\phi(\xi) \approx I_t^\alpha[{}^C D_t^\alpha u](\xi)$$ (for fractional dynamics)

Encoder Choices

• MLP encoder: three hidden layers, 33 neurons per layer with $\tanh$ activation
• KAN encoder: three hidden layers, 15 neurons, each neuron is a tunable Chebyshev-polynomial map (degree $k=4$), with $\tanh$ nonlinearities between layers

KANs simultaneously learn linear weights and coefficients of univariate Chebyshev activation maps, yielding enhanced parameter efficiency for nonlocal operator learning.

Explicit Solution Reconstruction

Given $M_\theta$ (and $J_\phi$ in the fractional case), MINPO reconstructs the solution $u$ via an explicit differentiable ansatz:

$$u_\Theta(\xi) = u_0(x) + \sum_{|\gamma|\geq 0} a_\gamma(\xi)\,\nabla_\xi^\gamma M_\theta(\xi) + \tilde{V}(\xi;\Theta)$$

• $\gamma$ indexes spatial/time derivatives $\nabla_\xi^\gamma$
• Coefficients $a_\gamma(\xi)$ arise from applying Leibniz' rule (or corresponding identities for fractional inversion)
• $\tilde{V}(\xi;\Theta)$ enforces initial traces or fractional inverse requirements

Common reduction cases:

Experiment	Solution Reconstruction
Volterra IDE (Exp I)	$u(t) = \partial_t M_\theta(t) + M_\theta(t)$
3D nested IDE (Exp II)	$$u(x_1,x_2,t) = \partial^3_{t,x_1,x_2} M_\theta + \partial^2_{x_1,x_2} M_\theta$$
Fractional PDE (Exp III)	$u(x,t) = u(x,0) + J_\phi(x,t)$

3. Loss Functions and Training Procedure

All neural parameters $\Theta = (\theta, \phi)$ are jointly optimized using a composite loss function:

$$L(\Theta) = \lambda_{\text{IDE}} L_{\text{IDE}} + \lambda_{\text{data}} L_{\text{data}} + \lambda_M L_M$$

• Physics-residual loss ($L_{\text{IDE}}$): measures MSE residual over collocation points,

$$L_{\text{IDE}} = \text{MSE} \left[ T_\alpha[u_\Theta] - N[u_\Theta] - M_\theta - S \right]$$

Crucially, no kernel or fractional term is discretized in this residual; all nonlocality resides in $M_\theta$ and $J_\phi$.

• Data-fidelity loss ($L_{\text{data}}$): penalizes deviation from observed or boundary values,

$$L_{\text{data}} = \text{MSE}[u_\Theta - u_{\text{obs}}]$$

• Nonlocal-consistency loss ($L_M$): enforces agreement between learned memory field and its integral representation,

$$L_M = \text{MSE} \left[ M_\theta(\xi_i) - \int_{\Omega_{\xi_i^-}} K(\xi_i, \eta; u_\Theta(\eta)) d\eta \right]$$

For fractional regimes:

$$L_M = \text{MSE} \left[ M_\theta(x, t) - \frac{1}{\Gamma(1-\alpha)} \int_0^t \partial_\tau u_\Theta(x, \tau) (t - \tau)^{-\alpha} d\tau \right]$$

These integrals are computed only inside $L_M$ via low-cost quadrature (e.g., Gauss–Legendre).

4. Operator Inversion and Solution Recovery

MINPO exploits the injectivity and analytic properties of the memory operator $M[u]$ (with known kernel type), ensuring reconstruction via closed-form identities derived from Leibniz’ rule or fractional-calculus relations. Once $M_\theta$ (and $J_\phi$ for $\alpha < 1$) are learned, $u_\Theta$ is recovered by differentiating $M_\theta$ and composing with learned inverse, without need for iterative inversion, history tracking, or explicit quadrature during inference.

This explicit treatment distinguishes MINPO from solvers that require repeated evaluation or accumulation of nonlocal integrals.

5. Empirical Evaluation Across Representative Nonlocal Problems

MINPO has been quantitatively benchmarked against classical spectral/fd solvers, A-PINN/fPINN, and their newly-introduced KAN analogues (A-PIKAN/fPIKAN).

Experiment I: Nonlinear Volterra IDE (1D, Forward & Inverse)

• Equation: $$\displaystyle \frac{du}{dt} + u = \kappa \int_0^t e^{\tau-t} u(\tau) d\tau$$; analytic solution $u(t)=e^{-t}\cosh(\sqrt{\kappa}\, t)$
• Forward ($\kappa$ known): $N_{\text{IDE}}=2400$
  • MINPO-KAN ($\kappa=1$): $E(u)\approx1.49\times10^{-4}$, $E(M)\approx1.89\times10^{-4}$
  • A-PIKAN: $3.27\times10^{-4}/2.85\times10^{-4}$,
  • A-PINN: $4.28\times10^{-4}/2.22\times10^{-4}$
  • MINPO-KAN reduces solution error by $35$–$45\%$ (vs. A-PIKAN), $55$–$65\%$ (vs. A-PINN)
• Inverse ($u, \kappa$ unknown, 10 measurements):
  • MINPO-KAN: $\kappa$ error $O(10^{-5}$–$10^{-4})$, substantially lower solution/operator errors than baselines

Experiment II: 3D Nested Nonlocal IDE

• Equation: $$(\partial_t+\partial_{x_1}+\partial_{x_2})u = u + \int_0^{x_2}\int_0^{x_1}\int_0^t e^{\tau-t} u(y_1,y_2,\tau) d\tau dy_1 dy_2 + f(\xi)$$
• Boundary constraint for $M$ is hard-wired into architecture
• MINPO-KAN: $E(u)=4.49\times10^{-3}$, $E(M)=1.46\times10^{-3}$ (Table 3)
  • 76.7% u-error reduction and 88.9% M-error reduction, $4.7\times$ faster vs. finite-difference baseline
  • MINPO-MLP achieves up to $15.3\times$ acceleration

Experiment III: 1D Time-Fractional Diffusion

• Equation: $${}^C D_t^\alpha u(x,t) = \partial_x^2 u(x,t) + S(x,t)$$, Dirichlet initial/boundary data
• MINPO computes $u(x,t)=u(x,0)+J_\phi(x,t)$
• For coarse $N_t=10$, MINPO-KAN $u$-error $\approx0.7$–$1.8\%$, $M$-error $\approx2$–$6\%$, whereas fPIKAN/fPINN $u$-error $\approx1.8$–$3.5\%$, $M$-error $\approx10$–$18\%$
• MINPO yields uniformly improved accuracy as $N_t$ increases and smoother memory operator fields

Summary Performance Table

Experiment	MINPO-KAN Accuracy	Competing Method Accuracy	Speedup
Volterra IDE	$E(u)\approx1.49\times10^{-4}$	$3.27\times10^{-4}$ (A-PIKAN)	—
3D Nested IDE	$E(u)=4.49\times10^{-3}$	$8.98\times10^{-3}$	$4.7\times$
Fractional PDE	$u$-error $0.7$–$1.8\%$	$1.8$–$3.5\%$	—

6. Generalization Properties and Prospective Applications

MINPO generalizes across IDE classes due to:

• Continuous enforcement of the governing physics; neither discretized quadrature nor collocation are required for nonlocal/fractional terms in the physics residual
• The nonlocal-consistency loss, computed via a single low-cost quadrature, acts as a regularizer independent of physics enforcement
• Explicit solution reconstruction, eliminating the need for cumulative history or discretization artifacts
• Flexibility to trade expressivity and parameter count between KANs and MLPs

Scalability

MINPO demonstrates robust performance in 1D, nested 3D, and fractional IDE settings. It avoids the curse of kernel dimensionality by having the physics residual term free of volumetric kernel summation, and achieves accuracy with orders-of-magnitude fewer degrees of freedom than finite-difference methods.

Potential Extensions

MINPO offers a framework suited to:

• Multi-term/distributed-order fractional PDEs
• Spatial fractional Laplacians, tempered Lévy kernels, peridynamics
• State-dependent or time-varying memory kernels (applications in hysteresis or adaptive systems)
• Integration with hybrid symbolic-neural kernel discovery via sparse regression

MINPO provides a unified framework to learn and solve IDEs of classical, fractional, spatially nonlocal, or mixed types by representing the nonlocal operator and its inverse as neural fields, enforcing the physics continuously, and reconstructing the solution via operator-theoretic identities. Across benchmark tasks, MINPO consistently outperforms classical solvers, A-PINN/fPINN, and their KAN variants, achieving up to an order-of-magnitude increase in operator accuracy and computational speedup of $5$–$15\times$ (Mostajeran et al., 19 Dec 2025).