Computer Science and Mathematics

Abstract: Physical infrastructure failures expose a structural blind spot in conventional financial risk systems: when the electricity supply becomes binding, markets enter metabolic arrest, a state where information-processing capacity collapses, and standard correlation-based metrics misread rising cross-asset correlations as stability. This paper introduces TENSORnet (Temporal Entropy-aware Network for Systemic Onset Recognition), a physics-informed computational protocol that fuses a VIX-calibrated inhibitory gate γL(t) (NLS-fitted, R2 = 0.30) with cross-asset Shannon entropy to detect infrastructure-induced stress before it materialises in price-based indicators. Applied to a temporal cross-asset graph of 2,838 JSE trading days (05 January 2015 to 29 April 2026) across seven asset classes under South Africa’s load-shedding crisis, TENSORnet achieves Precision = 100.0%, Recall = 85.8%, F1 = 92.4%, AUC = 1.000, and zero false positive alarms (Stage 3+ definition) on 1,830 out- of-sample days, with a mean lead time of 17 calendar days (408 hours) before fragile regime onset, outperforming all benchmarks, including XGBoost (F1 = 71.3%). Ablation confirms the physics-informed gate as the dominant architectural component (ΔF1 = −92.4 pp on removal): statistical learning alone recovers nothing (AUC = 0.469). The Densification Paradox, rising cross-asset correlation with falling entropy under stress (r = −0.468, p < 0.001), is confirmed empirically for the first time in cross-asset data. A Thermodynamic Port-Hamiltonian Neural ODE (TPH-NODE) extension grounds metabolic arrest in the second law of thermodynamics.

Abstract: Differential Causal Networks (DCNs) were introduced to represent changes between two causal networks inferred under different conditions. In their original use, however, DCNs remain pairwise objects: each differential graph summarizes rewiring within a single system, while common differential structures shared across many systems remain implicit. We introduce a methodological framework for the local alignment of DCNs aimed at detecting recurrent rewiring motifs, that is, small directed differential subnetworks that reappear across multiple systems under the same contrast. The proposed framework transforms each system-specific comparison into a signed directed differential graph, preserves both edge direction and change type, and searches for approximate local correspondences rather than enforcing a full-network mapping. The method consists of four steps: construction of signed DCNs, extraction of differential seeds, pairwise local alignment by seed-and-extend, and progressive multiple alignment to build consensus motifs. We define a score that combines node compatibility, differential-edge conservation, directional consistency, and recurrence support, and we complement the alignment procedure with null-model testing and robustness analysis. The result is a collection of consensus local differential modules ranked by recurrence, confidence, and statistical significance. In this formulation, DCNs become comparable units in a higher-order analysis whose goal is not merely to describe pairwise causal change, but to identify the same local rewiring logic reused across multiple systems.

Abstract: We present a unified, self-contained analytic treatment of the restricted weighted Goldbach representation function R_a,q(N) := ∑_{p₁ + p₂ = N, p₁ ≡ a (mod q)} (log p₁)(log p₂), q ≥ 1, (a,q) = 1,and its ternary analogue W_a,q(n) := ∑_{p₁ + p₂ + p₃ = n, p₁ ≡ a (mod q)} (log p₁)(log p₂)(log p₃). The binary theory is organized into three analytic levels: Level 1 (unconditional almost-all theorem with effective constants K ≤ 3.3624); Level 2 (valid for all sufficiently large N under a zero-density hypothesis); Level 3 (GRH-conditional pointwise asymptotic with explicit threshold N₀(4) ≈ 10^19.9). We incorporate four structural corrections over previous versions: (C1) replacement of an invalid pointwise Weyl–Pólya–Vinogradov bound by a rigorous appeal to Iwaniec–Kowalski; (C2) replacement of a misapplied hybrid large sieve by Parseval's identity; (C3) a parameter-compatibility lemma closing the gap in the arbitrary-A minor-arc saving; (C4) a corrected second-moment derivation for the restricted error R_a,q(N) – M_a,q(N) via the character decomposition. Beyond the corrections, we prove three new results: (N1) a Chen-type theorem giving N = p + P₂ with p ≡ a (mod q) for every sufficiently large even N; (N2) a short-interval theorem guaranteeing R_a,q(n) > 0 in every interval [N, N + N^0.525]; (N3) an analytic bridge from the explicit formula for Ψ*(x) deriving the precise reason why the Mellin transform of the residuals ε(p) detects the imaginary parts of the non-trivial zeros of ζ(s). We also present a rigorous three-level ternary hierarchy via prime anchoring and a completed ternary singular series analysis for q = 4. A complete table of effective constants with their epistemic status is provided, and the paper lists four precisely formulated open problems. Riemann Hypothesis, effective constants, Siegel zeros, ternary Goldbach, singular series, zero-density estimates, Mellin transform, spectral analysis, Riemann zeta function.

Abstract: Optimisation algorithms play an important role in the solution of nonlinear engineering design problems, particularly where objective functions exhibit complex, nonconvex, and potentially multimodal behaviour. Classical gradient-based methods, including the Method of Moving Asymptotes (MMA) and Sequential Quadratic Programming (SQP), are widely recognised for their computational efficiency and rapid local convergence; however, their performance may be sensitive to the presence of local extrema. In contrast, metaheuristic approaches such as Particle Swarm Optimisation (PSO) generally provide enhanced global exploration capabilities, albeit often at significantly greater computational expense. This study presents a preliminary investigation of a hybrid optimisation framework termed the Constrained Adaptive Model-based Exploration Optimiser (CAMEO). The proposed approach combines bounded stochastic exploration with constrained local refinement in an attempt to improve robustness within multimodal optimisation landscapes whilst retaining the efficiency associated with deterministic optimisation methods. The performance of the proposed framework was examined using a series of benchmark optimisation problems and compared against MMA, SQP, and PSO. The numerical results indicate that CAMEO is capable of attaining solutions closer to the global optimum in several test cases, whilst maintaining stable convergence characteristics.

Abstract: Modern machine learning systems can achieve remarkable predictive performance. Nevertheless, in several fields, this is not enough to produce acceptable solutions as we need formal guarantees of robustness, fairness, and interpretability. Most existing approaches treat these properties separately or introduce them through external constraints, which makes their interaction difficult to analyze. In this work, we develop a unified variational perspective that incorporates these requirements directly into the learning objective. Concretely, we model learning as the minimization of a composite functional that combines predictive risk, regularization, and additional terms that capture robustness, fairness, and interpretability. This viewpoint allows us to study these properties within a single mathematical framework. Under standard assumptions, we prove the existence of minimizers and show that the resulting solutions are Pareto-optimal for the associated multi-objective problem. We illustrate the framework using examples based on adversarial and distributional robustness, statistical fairness criteria, and a notion of interpretability. The analysis points out the trade-offs that inevitably arise. We also examine statistical aspects of the proposed objective and show that classical generalization guarantees can still be obtained under appropriate conditions. The resulting framework provides a flexible basis for designing reliable learning systems.

Abstract: We present and analyze a new weighted family of iterative methods for solving systems of nonlinear equations. The proposed schemes are constructed as a generalization of the fifth-order method of Singh et al. by incorporating appropriate weight functions into the correction step, thereby generating a flexible class of methods that includes the original scheme as a special case. Sufficient conditions on the weight functions are established to guarantee fifth-order local convergence. Several admissible choices are presented to illustrate the versatility of the family. The practical performance of the proposed variants is investigated on a collection of large-scale nonlinear systems. Furthermore, the family is applied to the nonlinear algebraic system obtained from the finite-difference discretization of a stationary one-dimensional viscous Burgers problem. Numerical experiments indicate that the proposed methods provide a competitive and accurate alternative for solving nonlinear systems of this type.

Abstract: This paper develops a robust numerical scheme based on a frame collocation method for solving multi-term fractional ordinary differential equations (FODEs) whose solutions exhibit multiple singularities at the origin. To adaptively capture the singular behavior, we construct a hybrid basis function frame by combining shifted fractional Legendre polynomials. An efficient computational formula for the Caputo fractional derivative is derived, which transforms the original problem into a nonlinear algebraic system at the collocation points. The resulting system matrix is severely ill-conditioned due to the redundancy of frame, to mitigate this issue, we employ truncated singular value decomposition (TSVD) regularization, thereby enabling stable and high-precision solutions. Extensive numerical experiments on several benchmark problems, including the fractional Bagley–Torvik equation, linear multi-term FODEs, and nonlinear cases, demonstrate that the proposed method achieves exponential convergence rates. Notably, when the singular exponent of the solution matches a tunable parameter $\delta$ in the basis functions, superconvergence is observed, significantly outperforming standard spectral methods. Compared with traditional spectral approaches, the proposed frame collocation framework retains spectral accuracy while exhibiting superior capability in handling complex singular structures, providing a powerful and reliable tool for high-precision simulations of multi-term fractional differential equations.

Abstract: APD is widely adopted in the management of end-stage renal disease (ESRD) and offers flexi-bility and improved quality of life, but bacterial infections, particularly peritonitis, are still a major constraint, which frequently results in hospitalization, catheter failure, and hemodialysis. Early diagnosis is important but difficult because of the non-specific clinical manifestations and delays related to the traditional diagnostic techniques like culture-based analysis. “To overcome these restrictions, this paper suggests a new explainable machine learning model to early identify bacterial infections in APD patients based on multimodal data streams, such as clinical, lab, and time-series dialysis data, to identify both fixed and dynamic infection onset patterns”. The framework uses a hybrid characteristic of feature engineering, which is a combination of statistical selection techniques and clinically relevant indicators to improve predictive performance, and Supervised learning models of high accuracy like the Random Forest, SVM, and Gradient Boosting are applied. One of the contributions of this work is the incorporation of explainable artificial intelligence through SHAP that leads to a clear interpretation of model predictions and the determination of key risk factors that will affect the development of the infection and thus enhance clinical trust and usability. The experimental findings indicate that the given approach greatly enhances the accuracy of early detection as compared to the conventional ones, allowing timely intervention, minimizing complications, and improving the overall outcomes of the treatment, which underscores its potential as a scalable and clinically applicable decision support system to manage APD.

Abstract: We develop a fast Chebyshev spectral collocation method for a coupled system of nonlinear Klein–Gordon equations augmented by Caputo-type fractional memory integrals. The governing equations retain the classical second-order time derivative as the leading operator and incorporate weakly singular convolution integrals with power-law kernels t−α, α∈(0,1), modelling viscoelastic memory damping rather than replacing the wave operator. The spatial discretisation employs Chebyshev–Gauss–Lobatto collocation, while the temporal integration uses a Newmark scheme (βNM=1/4) with the spatial operator treated implicitly and both the L1 memory sums and the cubic nonlinearities evaluated explicitly at the known time level; a linear extrapolation of the nonlinear terms eliminates the need for Newton–Raphson iterations. The disparate memory tails arising from two distinct fractional orders α≠β are compressed by independent Sum-of-Exponentials (SOE) approximations, reducing the per-step memory cost from O(Nt) to O(p+Nexp) and the total complexity from O(Nt2) to O(Nt(p+Nexp)). A rigorous stability estimate and a global convergence bound are established using a discrete Gronwall inequality. Numerical experiments confirm the temporal convergence rate O(Δtmin(2−α,2−β)), spectral spatial accuracy, and the practical speedup afforded by the SOE acceleration. A solitary wave collision scenario illustrates the method’s capability to capture asymmetric dispersive wakes generated by the fractional memory. The algorithmic architecture is dimension-independent by construction; a concrete extension pathway to multi-dimensional tensor-product Chebyshev grids, including Kronecker-product operators and Sylvester-based solvers, is presented.

Abstract: This paper consolidates, corrects, and extends a research programme on the shifted-prime problem $p = q + r - 1$ with $p, q, r$ prime and its connections to the binary Goldbach conjecture and the non-trivial zeros of the Riemann zeta function $\zeta(s)$. New material over Version 6. The principal addition is a rigorous three-level treatment of the restricted Goldbach sum \[ P_{R_{3,4}}(N) = \sum_{\substack{p+q=N,\\ p\equiv 3\ (\mathrm{mod}\,4)}} (\log p)(\log q). \] At Level 1 [PROVED] (unconditional), the ``almost-all'' theorem of Montgomery--Vaughan type shows that the exceptional set of even integers $N\leq X$ for which $|R_{3,4}(N) - \tfrac{1}{2}C_2 S(N)N|$ exceeds $CN/(\log N)^3$ has measure $O_A\bigl(X/(\log X)^A\bigr)$ for every $A>0$. At Level 2 [PROVED] (unconditional), a transfer inequality bounds $|R_{a,q}(N)-\phi(q)^{-1}R(N)|$ in terms of twisted sums $S_\chi(N)$ with mean-square control. At Level 3 [COND. PROVED, GRH], for all sufficiently large even $N$ one has $R_{3,4}(N)=\tfrac{1}{2}C_2 S(N)N + O(N^{1/2+eps})$. Anderson's original claim of an explicit unconditional constant $K\leq 28.65$ for all $N$ is identified as relying on the Hardy--Littlewood binary asymptotic for each individual $N$, which is itself a conjecture; the claim is accordingly downgraded and the gap stated precisely. Retained from Version 6. Five analytical gaps (A--E) in the Goldbach--Riemann bridge for $\Psi^*(x)$ are fully closed unconditionally (Gaps D1, D2, D3, E) or under GRH (Gap C). The corrected spectral-detection results stand: $\lambda_1/\lambda_2 = 182.63$ ($n=892\,206$); 129/200 Riemann zeros detected at $p<0.01$ ($n=1\,310\,763$); Mellin--Lomb--Scargle concordance 29/30 versus 0/30; 9/10 direct Pearson correlations significant; heteroscedasticity of $eps(p)$ formally confirmed ($p=4.7\times 10^{-14}$). Principal corrections retained from Version 6. The $k=3$ existence problem is equivalent to binary Goldbach (open). The permutation-test bug in scripts~6.py--8.py is corrected ($199/200\to 129/200$). The formula for $S_\infty^{(k)}$ is corrected for $k\geq 3$. None of these results constitutes a proof of the Riemann Hypothesis. All claims carry explicit epistemic labels.

Abstract: Numerical simulation of the multi-dimensional space-fractional Cahn-Hilliard equation faces two main computational challenges: the inherent temporal accuracy limitations of standard scalar auxiliary variable (SAV) methods and the escalating computational cost in high-dimensional domains. To address these issues, this study constructs a fully discrete algorithmic framework integrating a second-order backward differentiation formula (SAV-BDF2) with a sixth-order centered difference scheme. Under this formulation, we rigorously prove unconditional energy stability and establish the theoretical validity of the dual temporal and spatial accuracy. To solve the resulting indefinite algebraic systems, a minimal residual solver is paired with a sine-transform block diagonal preconditioner. Additionally, a hardware-level Vectorized Tensor Processing (VTP) architecture is deployed to resolve cache thrashing caused by non-contiguous memory access during multidimensional tensor evaluations. Numerical experiments in 3D to 8D domains demonstrate that the framework improves memory throughput and reduces execution time. By avoiding standard hardware execution inefficiencies, this integrated strategy provides an efficient numerical solution for large-scale simulations of high-dimensional fractional systems.

Abstract: The Taylor Green vortex is a classic benchmark for unsteady incompressible flows, but most existing physics-informed neural network (PINN) studies on this problem report only early time results or omit the pressure field entirely. We present a complete, step by step PINN simulation of the two-dimensional decaying vortex at Reynolds number 100 using the DeepXDE library. A fully connected network with four hidden layers of 128 neurons each is trained on 20,000 collocation points, 2,000 boundary points, and 2,000 initial points for 10,000 Adam iterations. The entire training takes about 13 minutes on a single GPU. Relative $L_2$ errors for the velocity components $u$ and $v$ increase from approximately 5\% at $t = 0.25$ to 18\% at $t = 1.0$. Vorticity fields are captured qualitatively, but peak values are smoothed over time. All visualisations contour plots, line cuts, three dimensional surfaces, and the loss history are generated automatically from the trained model. This work provides a reproducible benchmark for researchers developing or testing PINN methods for unsteady fluid dynamics, and it openly discusses both the successes and the persistent difficulties of the approach.

Abstract: We describe a decimal–hexadecimal block encoding for primality over a finite stored range. Since every prime greater than 5 must lie in one of the residue classes 1, 3, 7, 9 (mod 10), each decimal block of size ten can be encoded by a 4-bit word indicating which of the candidates 10k + 1, 10k + 3, 10k + 7, and 10k + 9 are prime. This yields a nibble-based storage scheme supporting exact primality queries and exact recovery of the prime-counting function π(x) by cumulative popcount. We then establish a structural theorem arising from the congruence 10 ≡ 1 (mod 3): for k ≡ 0 (mod 3) the candidates 10k + 3 and 10k + 9 are always composite, and for k ≡ 2 (mod 3) the candidates 10k + 1 and 10k + 7 are always composite. This partitions the nibble alphabet into three classes of sizes 4, 16, and 4, reducing the Shannon entropy from 4 bits to 2.42 bits per nibble and yielding a lossless compression of 39.4% over the original encoding with O(1) decode complexity. We present data structures, Python routines, and experimental validation up to 300,000.

Abstract: The numerical simulation of incompressible viscous flows remains a central pillar of modern computational fluid dynamics (CFD). Over the past decades, a wide spectrum of numerical methodologies has been developed, reflecting fundamentally different mathematical formulations and discretization philosophies. Among these, domain-based approaches—such as finite difference, finite element, finite volume, and meshfree methods—have emerged as versatile and general-purpose frameworks, while boundary element methods provide efficient alternatives for problems governed by linear physics, particularly in unbounded domains. This review presents a comprehensive examination of the historical development, mathematical foundations, and computational characteristics of these approaches for Newtonian incompressible flows. Emphasis is placed on the conceptual distinctions between boundary-integral and domain-based formulations, their applicability to internal and external flow regimes, and their compatibility with turbulence modeling strategies, including Reynolds-averaged Navier–Stokes (RANS), large-eddy simulation (LES), and direct numerical simulation (DNS). The intention is to provide a unified perspective that clarifies the strengths and limitations of the principal CFD methodologies and offers guidance on their suitability for different classes of flow problems.

Abstract: Cryptographic security proofs are the invisible backbone of modern digital systems, yet they remain fragmented across multiple paradigms—game-based proofs, Universal Composability (UC), formal verification, and ad hoc insecurity arguments—each with its own language, assumptions, and limitations. This article introduces the \textbf{Market-Theoretic Security Framework (MTSF)}, a unified paradigm that reinterprets all security proofs as economic markets. In this view, the defender acts as a seller offering \emph{security goods} (such as confidentiality or unforgeability), while the adversary acts as a buyer bidding computational resources to break them. Security emerges naturally as \emph{market equilibrium}, where no efficient adversary can afford to win, while insecurity is characterized as \emph{market collapse}, where attacks succeed at negligible cost. For cryptographers, MTSF provides a rigorous and expressive framework that unifies four major proof paradigms into a single formal language. It introduces key technical innovations such as the \textbf{extended difference lemma} for handling multiple simultaneous failure events, \textbf{bidding-based reductions} that explicitly model adversarial strategies, a \textbf{dual methodology that treats proofs and disproofs symmetrically within the same structure}, and a \textbf{session pinging mechanism} for unbounded session verification. The framework seamlessly extends to classical and post-quantum primitives, real-world protocols (including TLS~1.3 and Signal), and even quantum-adversarial settings, while preserving quantitative security bounds and composability guarantees.MTSF offers an intuitive, accessible, and powerful meta model: security is like a marketplace where attackers try to ``buy'' a break, and defenders ensure the price is prohibitively high. Each proof becomes a sequence of small price adjustments, and each attack corresponds to a failed or successful bid. By combining mathematical rigor with economic intuition, MTSF transforms security proofs from opaque technical artifacts into transparent, auditable, and universally understandable arguments, enabling both experts and practitioners to reason about security with clarity and confidence.

Abstract: In this paper, we consider the properties of the following objects: plafal and geo-space (a general overview). As an application of the created theory, the proof of the equality of complexity classes P and NP will be given. The geo-plafal is a kernel (computational template) of the proof; constructive theory of serendipity approximations, Stepanets' school and the Bogolyubov principle of the decay of correlations for an infinite systems (dim=3) is a shell.

Abstract: Insurance pricing plays a central role in risk management and financial decision-making, 2 as accurate premium estimation directly impacts portfolio stability and profitability. This 3 study investigates insurance pure premium estimation by integrating classical actuar- 4 ial models with modern machine learning techniques. We compare the traditional fre- 5 quency–severity decomposition framework with direct modeling approaches, including 6 XGBoost and Tweedie models. For claim frequency, we evaluate Poisson-based models, 7 generalized additive models, and XGBoost. For claim severity, we compare a Gamma gen- 8 eralized linear model with XGBoost. The results show that XGBoost significantly improves 9 predictive performance for both components. Within the decomposition framework, the 10 XGBoost–XGBoost model achieves the best overall prediction accuracy. However, lift-based 11 analysis reveals that the XGBoost–Gamma model provides superior risk segmentation, 12 highlighting a trade-off between prediction accuracy and risk ranking. Direct modeling 13 approaches, while competitive, do not outperform the decomposition framework. Overall, 14 the findings demonstrate that machine learning enhances predictive performance, but its 15 effectiveness is maximized within the frequency–severity framework. The results further 16 indicate that claim frequency is the primary driver of risk differentiation, while claim sever- 17 ity contributes more to prediction accuracy. These findings have important implications for 18 risk management and pricing strategies in insurance portfolios.

Abstract: Natural rubber price forecasting is inherently difficult due to nonlinear, non-stationary dynamics driven by supply fundamentals, cross-market signals, exchange rate movements, and speculative trading. This study proposes VMD–Hybrid BiLSTM–Transformer, a dual-pathway framework integrating Variational Mode Decomposition (VMD) with a Bidirectional LSTM encoder and a Transformer encoder for daily RSS3 FOB price change forecasting. Rather than forecasting each intrinsic mode function independently, all five VMD components are appended directly to the economic feature matrix — preserving multi-scale frequency information within a single forward pass and avoiding the variance collapse observed in conventional decomposition forecast approaches (StdR = 0.04 - 0.15). On a 237-observation held-out test set (September 2025–February 2026), the model achieves Pearson correlation of 0.812, directional accuracy of 67.1%, and StdR of 0.819, outperforming ARIMA by 0.662 in correlation and 37.3% in MAE, with predictive skill confirmed up to five days. These results demonstrate that directional accuracy alone is insufficient for evaluating difference commodity price models, and that jointly integrating multi-scale decomposition, bidirectional learning, and global attention is essential for reliable agricultural price forecasting.

Abstract: The flow past two cylinders in tandem arrangement is of fundamental importance in engineering applications such as heat exchangers, offshore structures, and power transmission lines. This study presents a complete open‑source simulation pipeline using Gmsh for mesh generation and OpenFOAM for the finite‑volume solver, combined with a long short‑term memory (LSTM) neural network surrogate for fast predictions. A distance‑based refinement strategy resolves the flow accurately, with characteristic mesh sizes as low as around the cylinders. The methodology is validated against the classical Schäfer–Turek single‑cylinder benchmark at (Re=100), showing satisfactory agreement for force coefficients and Strouhal number. The main analysis focuses on a tandem configuration at \( Re=1.0\times10^5 \) with unequal diameters (\( D_1=0.1\;\mathrm{m} \), \( D_2=0.15\;\mathrm{m} \)) spaced \( 1.0\;\mathrm{m} \) centre‑to‑centre. The results reveal strong wake interaction: the downstream cylinder experiences higher mean drag \( (\overline{C}_D=0.997) \) and significantly larger lift fluctuations \( (C_L'=0.340) \) than the upstream cylinder \( (\overline{C}_D=0.947 \), \( C_L'=0.129 \)). Both cylinders shed vortices at the same frequency \( f=2.041\;\mathrm{Hz} \), yielding Strouhal numbers \( St_A=0.204 \) and \( St_B=0.306 \). An LSTM neural network trained on the force coefficient time series achieves near‑perfect predictions of the downstream lift and correctly reproduces the shedding frequency, providing a fast and accurate surrogate model. The fully reproducible open‑source workflow, including all CFD setup files and the neural network training code, is made available, enabling future studies on bluff‑body interactions and facilitating the adoption of data‑driven methods in fluid mechanics.

Abstract: This paper applies physics-informed neural networks (PINNs) to laterally excited liquid sloshing in a two-dimensional rectangular tank, where near-resonant forcing (ω_e/ω₁ = 0.9) produces a multi frequency beating response with a period of approximately 10T₁. Linearised potential flow theory governs the problem; the network learns the velocity potential φ(x,z,t) while the free-surface elevation η is injected analytically. Two training obstacles specific to forced sloshing are analysed. First, a zero solution trap arises because the trivial solution φ̂=0 satisfies all equations except the free-surface conditions, whose residuals are roughly 10⁻⁴ times smaller than the Laplace residual; characteristic scale normalisation combined with loss weighting (λD = λK = 100) breaks this trap. Second, spectral bias prevents standard MLPs from resolving the three co-existing frequencies (ω₁, ω_e, ∆ω); a Fourier time embedding that augments the input from 3 to 9 dimensions overcomes this limitation. Two additional techniques further reduce errors: a hard wall boundary condition enforced exactly via a cos(πx/B) spatial embedding, which eliminates wall collocation points; and a gradient-enhanced Laplace regulariser (∥∇(∇²φ̂)∥²) that constrains velocity smoothness through third-order automatic differentiation. An ablation study shows that these four techniques progressively reduce the horizontal velocity error from ε_u = 12.46% to 0.84%. Results are validated against a viscous finite-difference benchmark. Over one beating cycle the errors are ε_η = 0.15%, ε_u = 0.84%, and ε_w = 1.65%. Afrequency parameter study across ω_e/ω₁ = 0.5–1.1 gives ε_η < 0.25% and ε_u < 2.3% for all near-resonance cases. For long-time simulation, a time-domain decomposition strategy with transfer learning partitions the domain into one-beat windows; extending to five beating cycles (50T₁) yields ε_u = 3.43% and ε_η = 0.30% with no monotonic error accumulation across windows. The methodology is then extended to a three-dimensional rectangular tank (B × W × H) with bi-directional lateral excitation. The 3-D formulation introduces the y-dimension into the Laplace equation (∇²φ = φ_xx + φ_yy + φ_zz = 0), adds transverse wall boundary conditions (∂_φ/∂_y = 0) enforced exactly via a cos(πy/W) embedding, and extends the Fourier time embedding from 9 to 16 dimensions to accommodate six physical frequencies. The bi-directional excitation excites both (m,0) and (0,n) modal families, producing a genuinely three dimensional beating response. Results demonstrate that the proposed techniques transfer effectively to 3-D, with errors ε_η = 0.24%, ε_u = 1.31%, ε_v = 1.78%, and ε_w = 2.32% over one beating cycle (2,499 s training time).