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REVIEW 3 major objections 4 minor 76 references

This paper argues that split complex-valued physics-informed neural networks solve forward and inverse nonlinear PDEs more accurately than real-valued ones.

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Split complex-valued PINNs achieve lower benchmark errors than real-valued PINNs, but the comparison is confounded by doubled parameters and unresolved internal error inconsistencies.

T0 review reviewed 2026-08-02 challenge →

load-bearing objection Broad benchmark suite for split-complex PINNs, but the headline claim is not yet supported: the comparison against real PINNs is confounded by parameter count, and the reported Burgers error does not reproduce in the ablation tables. the 3 major comments →

arxiv 2607.15087 v1 pith:UIZYF7AF submitted 2026-07-16 physics.flu-dyn

Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear PDEs

Biswanath Barman , Rajendra K. Ray , Debdeep Chatterjee This is my paper
classification physics.flu-dyn
keywords split complex-valued neural networkphysics-informed neural networksforward and inverse PDEsspectral biassplit activation functionssampling strategiesnonlinear Schrödinger equationBeltrami flow
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Physics-informed neural networks (PINNs) solve PDEs by training a network to satisfy the equation, but real-valued versions struggle with oscillatory, high-frequency, and phase-sensitive solutions. This paper proposes SCV-PINN, in which the network's weights and hidden activations are complex numbers, with standard real-valued activation functions applied separately to the real and imaginary parts. The physical solution is read from the real part, while the imaginary part serves as an auxiliary latent channel. Across a broad set of forward and inverse benchmarks — including Burgers, Allen–Cahn, Korteweg–de Vries, nonlinear Schrödinger, Helmholtz, Navier–Stokes flows, and a 3D Beltrami flow — the paper reports consistently lower relative L2 errors and better parameter recovery than real-valued PINNs. If correct, this offers a simple, general upgrade to PINNs for tough nonlinear dynamics.

Core claim

The central claim is that moving the trainable weights, biases, and latent features of a PINN into the complex domain — while keeping activations split (real functions applied to real and imaginary parts independently) — yields a strictly more expressive approximator that still trains with standard real-valued optimizers. The network output's real part is the physical solution; the imaginary part is a free auxiliary field that enriches the approximation without altering the physics loss. The authors show that this construction matches or beats conventional real-valued PINNs across a wide suite of nonlinear, oscillatory, and high-dimensional problems, with the 3D Beltrami flow reaching a rela

What carries the argument

The core object is the split complex-valued multilayer perceptron: weights and biases are complex, inputs are embedded as complex numbers with zero imaginary part, and the activation function is applied independently to the real and imaginary components (e.g., split GELU or split tanh). This preserves the amplitude–phase coupling of complex multiplication while avoiding holomorphic activation restrictions imposed by Liouville's theorem. The physical prediction is the real part of the output, so the imaginary part acts as an auxiliary latent channel. Training uses a split view: the optimizer operates on real and imaginary parts as real variables, with Adam followed by L-BFGS.

Load-bearing premise

The load-bearing premise is that the accuracy gains come from the complex-valued architecture itself, rather than from the roughly doubled number of trainable parameters or from the discarded imaginary part of the output acting as an unpenalized auxiliary variable.

What would settle it

Run a matched-parameter-count comparison: for each SCV-PINN, construct a real-valued MLP with the same number of real trainable parameters (e.g., double the hidden width) and train it with the same Adam/L-BFGS budget on the same collocation points. If that real network matches or beats SCV-PINN's relative L2 errors, the central claim is a parameter-count effect, not a complex-structure effect. A second check: freeze the imaginary part of the output to zero after initialization (so only real-part weights train) and see whether accuracy collapses; if it does not, the imaginary channel is not doi

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the reported accuracy holds, SCV-PINNs can be dropped into existing PINN pipelines as a direct replacement for real-valued networks without changing the loss formulation or optimizer.
  • The method extends naturally to intrinsically complex-valued equations such as nonlinear Schrödinger, where both real and imaginary parts of the solution are recovered simultaneously from the complex output.
  • Improved parameter identification for inverse Burgers and Navier–Stokes problems suggests the framework could be applied to data-driven discovery of unknown physical coefficients in experiments.
  • The high accuracy on the 3D Beltrami flow benchmark indicates that complex-valued representations may help with high-dimensional, smoothly oscillatory Navier–Stokes solutions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The imaginary channel is never used in the final prediction, so SCV-PINN could be seen as an adaptive augmentation of the function space; a natural test is whether a real-valued network with an explicit auxiliary output channel (not used in the loss) matches the gains.
  • Because the paper's ablation shows ReLU-type split activations stall under L-BFGS, future variants for second-order optimizers should stick to smooth activations; this may also explain why some complex-valued PINN efforts reported unstable training.
  • If the benefit is genuinely from complex multiplication's phase coupling, the approach could be combined with Fourier-feature embeddings to push further into the high-frequency regime, though this is a speculation beyond the present results.
  • The authors do not report wall-clock times or parameter counts per benchmark; a practical reader would want to know the computational cost per unit accuracy before adopting the method.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper proposes split complex-valued physics-informed neural networks (SCV-PINNs), in which weights, biases, and latent representations are complex-valued while activation functions are applied separately to the real and imaginary parts. The physical solution is recovered as the real part of the network output. The method is evaluated on a wide range of forward and inverse benchmarks: Burgers, Allen–Cahn, KdV, nonlinear Schrödinger, Helmholtz on regular/irregular domains, Poisson on regular/L-shaped domains, Kovasznay flow, lid-driven cavity flow, Lorenz system identification, inverse Burgers, inverse Navier–Stokes, and a 3D Beltrami flow. The central claim is that SCV-PINNs consistently achieve significantly lower relative L2 errors and better parameter identification than real-valued PINNs and existing variants. Extensive ablations of split activation functions and sampling strategies are also reported.

Significance. If the central claim is correct, SCV-PINN would be a broadly applicable and practically useful extension of standard PINNs, particularly for oscillatory and phase-dependent problems. The manuscript's strengths are the breadth of benchmarks, the systematic ablation of activations and collocation sampling, and the clear presentation of the method. However, the load-bearing comparison is currently confounded by a parameter-count effect, and several internal inconsistencies in the reported numerical values prevent the claims from being verified as stated. The paper does not provide code or any measure of run-to-run variability, which further limits reproducibility. With additional control experiments and correction of the inconsistencies, the contribution could become solid; in its present form the evidence for the central claim is not yet sufficient.

major comments (3)
  1. [Sec. 4.3, Eqs. (58)–(62); Sec. 4.4.3, Eqs. (72)–(73); Sec. 4.4.1, Eq. (67); Table 10] The 'identical architecture' comparisons in Table 10 are not parameter-matched. A complex hidden layer with m units stores 2m real-valued parameters, so an m-unit complex SCV-PINN has roughly twice as many trainable parameters as an m-unit real RV-PINN. Since the physical output is taken as the real part (Eq. 67), the imaginary branch is an auxiliary latent representation that approximately doubles representational capacity. No real-valued baseline with doubled width (2m units) and otherwise identical settings is reported anywhere. The central claim—that the improvement is due to the split-complex structure rather than to parameter count—is therefore not yet supported. A same-parameter-count real MLP control is essential.
  2. [Sec. 5.4; Table 1; Table 9; Table 10; Appendix B, Table 12] The Burgers forward problem is reported with a relative L2 error of 6.73e-5 in Table 1/Table 9 and Sec. 5.4, but Table 10 (under 'identical hyperparameters') lists a relative L2 error of 1.09e-2 for the same problem. Appendix B, Table 12, reports 1.244e-2 for LHS with CGELU and 1.090e-2 for random sampling, which matches the Table 10 value but not the headline 6.73e-5. The paper does not explain which configuration produces the headline error, and which configuration is used in the fair comparison. This is a direct, quantifiable inconsistency in the central numerical evidence and must be resolved.
  3. [Sec. 5.13, Eq. (170); Table 7; Fig. 24 caption] The 'true' parameters for the inverse Navier–Stokes problem are inconsistent. The text (Eq. 170) states the exact values are λ1=1.0, λ2=0.01. Table 7 reports the true values as 1.00137 and 0.01000, while Fig. 24's caption states the exact values are 1.00137 and 0.01007. Without a single consistent reference, the reported parameter-identification accuracy cannot be evaluated, and the comparison against the vanilla-PINN is ambiguous.
minor comments (4)
  1. [Sec. 5.1, Eq. (102); Sec. 5.5, Eq. (117)] Typos: '2,sech(x)' should read '2 sech(x)', and '0.0001, u_xx' should read '0.0001 u_xx'.
  2. [Tables 9 and 10] The same benchmark problems appear with different relative L2 errors in Table 9 and Table 10 (e.g., Kovasznay u: 8.20e-5 vs 1.22e-4; Helmholtz regular: 2.17e-4 vs 6.14e-3). The paper should clarify that these correspond to different training budgets or configurations, and should state which table supports the headline claims.
  3. [Appendices A and B] The selection of CTanh/CGELU and LHS as the default choices is based on ablations performed on the Burgers benchmark, which is also a headline problem. This is a form of selection bias on the evaluation benchmarks; the authors should acknowledge this and, ideally, validate on a separate tuning set.
  4. [Data Availability] The statement says data are available upon request but no code is provided. For a scientific machine learning paper, releasing training code at least for the main experiments would substantially improve reproducibility. Additionally, all results appear to be single-run; reporting mean±std over multiple seeds would strengthen the claim of 'consistent' improvement.

Circularity Check

0 steps flagged

No circular derivation found: the paper is an empirical architecture study with external benchmarks; the main comparisons are experimental, not definitionally forced.

full rationale

The paper's central claims are numerical: SCV-PINN achieves lower relative L2 errors on standard forward and inverse PDE benchmarks (Burgers, Allen–Cahn, NLS, Helmholtz, Kovasznay, cavity, Lorenz, inverse Navier–Stokes, 3D Beltrami flow). These are measured against closed-form or reference solutions, not derived from the method's definition. The physical output is taken as the real part of the complex network output (Eqs. 67 and 94), but this is a stated modeling choice rather than a hidden reuse of the target result. The comparisons with RV-PINN in Table 10 are genuine experiments, although the 'identical architecture' protocol leaves the complex network with roughly twice the real parameter count; that is a possible experimental confound and a correctness/interpretability concern, but it is not a circular reduction: the improved errors are not forced by construction, and the paper does not define SCV-PINN's advantage in terms of the benchmarks it then reports. The two self-citations ([61], [62]) are contextual related-work mentions and are not load-bearing for the main claim. The ablation-based choices of activation functions and sampling strategies (Appendices A/B) are hyperparameter selections made before the headline results, not fitted values disguised as predictions. No step in the paper equates a prediction with an input by definition, so the circularity score is 0.

Axiom & Free-Parameter Ledger

4 free parameters · 3 axioms · 1 invented entities

The central claim rests on the choice of per-benchmark architectures, training schedules, sampling counts, and unstated loss weights, plus the assumption that same-width complex vs real is a fair comparison. No code, seeds, or error bars are provided, so the contribution-to-parameter-count trade-off is unresolved.

free parameters (4)
  • Per-benchmark network architecture = e.g., Burgers 8x20 CTanh; NLS 5x100 CGELU; 3D Beltrami 2x40 CGELU
    Layer count, width, and activation are hand-selected per problem; no sensitivity analysis is given, and the best activation is chosen from the same benchmarks.
  • Per-benchmark training schedule = 5,000/5,000 to 10,000/20,000 Adam/L-BFGS iterations
    Iteration budgets vary across benchmarks, and the exact budget for headline errors (e.g., Burgers 6.73e-5) is not stated clearly.
  • Collocation point count and sampling strategy = e.g., 10,000 LHS (Burgers); 20,000 LHS (NLS/Allen-Cahn); 2,000 (Helmholtz hole)
    LHS was selected after an ablation (Appendix B); point counts vary by problem and affect the reported errors.
  • Loss weights λ_f, λ_u, λ_d = not reported
    Eq. (86) defines weighted losses, but the values of the weights are never given; presumably all set to 1, which would deserve a statement.
axioms (3)
  • ad hoc to paper A real-valued MLP with the same layer width is a fair control for SCV-PINN.
    Assumed in the 'fair comparison' tables, but complex weights double the parameter count; the assumption is questionable and never tested with a wider real baseline.
  • domain assumption The imaginary component enriches the approximation space without affecting the physical output.
    Eqs. (66)-(67) in Sec. 4.4.1 make this an auxiliary latent; no theory or ablation demonstrates that this latent augmentation improves generalization beyond adding capacity.
  • domain assumption A single run per configuration is representative.
    All reported errors are single trials; no seeds, variance, or uncertainty quantification are given.
invented entities (1)
  • Auxiliary imaginary latent representation no independent evidence
    purpose: Provides extra degrees of freedom and 'phase' information while the physical solution is read from the real part.
    No independent falsifiable handle is provided; the imagined 'phase' is just the imaginary component of a structured wider real network.

reviewed 2026-08-02 · how reviews work

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Cite this review

Pith. "Pith review of Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear PDEs." pith.science (2026). https://pith.science/paper/UIZYF7AF

@misc{pith2026260715087,
  author       = {Pith},
  title        = {Pith review of: Split Complex-Valued Physics-Informed Neural Networks for Forward and Inverse Nonlinear PDEs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UIZYF7AF}},
  note         = {Machine review of arXiv:2607.15087}
}
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read the original abstract

Physics-informed neural networks (PINNs) have emerged as a powerful framework for solving forward and inverse partial differential equations (PDEs), but conventional real-valued PINNs (RV-PINNs) often suffer from spectral bias, limited expressivity, and reduced accuracy for high-frequency, oscillatory, and phase-dependent dynamics. In this work, we propose a generalized split complex-valued physics-informed neural network (SCV-PINN), in which network parameters and latent representations are defined in the complex domain. The framework employs split complex-valued activation functions by independently applying standard real-valued activations to the real and imaginary components, providing numerical stability, computational efficiency, and improved approximation capability. This formulation enables simultaneous learning of amplitude and phase information, enhancing the representation of nonlinear and oscillatory systems. Extensive ablation studies evaluate different split activation functions and collocation sampling strategies. The proposed framework is validated on forward and inverse PDE benchmarks including Burgers, Allen-Cahn, Korteweg-de Vries, nonlinear Schrodinger, Helmholtz, Poisson, Kovasznay flow (Re = 20), lid-driven cavity flow (Re = 100), the Lorenz system, inverse Burgers, inverse Navier-Stokes (Re = 100), and a three-dimensional Navier-Stokes Beltrami flow. For the Beltrami benchmark, SCV-PINN achieves a relative L2 error of 4.07 x 10^-5. Numerical results consistently demonstrate lower relative L2 errors and more accurate parameter identification than RV-PINNs and several existing PINN variants. The proposed SCV-PINN provides a robust and generalized extension of standard PINNs for complex-valued, multiscale, oscillatory, high-dimensional, and real-valued nonlinear PDEs.

Figures

Figures reproduced from arXiv: 2607.15087 by Biswanath Barman, Debdeep Chatterjee, Rajendra K. Ray.

Figure 1
Figure 1. Figure 1: Geometrical interpretation of complex numbers and their enhanced representation capability for vector fields, oscillatory dynamics, and streamline structures arising in complex physical systems and nonlinear PDEs. 2.4. Physical Interpretation: Complex Representation of Fluid Streamlines A key motivation for adopting complex-valued formulations in physical modeling arises from their natural ability to repre… view at source ↗
Figure 2
Figure 2. Figure 2: Architectural comparison between conventional real-valued deep neural networks (RV-DNNs) and complex￾valued deep neural networks (CV-DNNs), highlighting the enhanced representation capability of complex-valued feature learning. 4.4. Split Complex-Valued Physics-Informed Neural Networks(SCV-PINN) Let Ω ⊂ ℝ𝑑 denote the spatial domain and let  = [0, 𝑇 ] represent the temporal interval. We consider a general … view at source ↗
Figure 3
Figure 3. Figure 3: General workflow of the proposed split complex-valued physics-informed neural network (SCV-PINN) framework for solving general nonlinear partial differential equations, including both forward solution prediction and inverse parameter identification problems. 4.4.7. Loss Function The total training objective combines physics, initial–boundary, and data constraints. Physics loss. Let 𝑓 = {(𝑥 𝑖 𝑓 , 𝑡𝑖 𝑓 )}𝑓… view at source ↗
Figure 4
Figure 4. Figure 4: Reference solution, SCV-PINN predicted solution, and corresponding absolute error contours for the nonlinear Schrödinger equation in terms of the modulus field |𝜓(𝑥, 𝑡)|. The proposed SCV-PINN achieves a relative 𝐿2 error of 8.45 × 10−5 [PITH_FULL_IMAGE:figures/full_fig_p022_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Reference and predicted solution comparison for the nonlinear Schrödinger equation at 𝑡 = 0.78 s for the real part 𝑢(𝑥, 𝑡), imaginary part 𝑣(𝑥, 𝑡), and modulus field |𝜓(𝑥, 𝑡)|. The proposed SCV-PINN accurately captures the complex-valued wave dynamics with close agreement to the reference solution. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 22 of 54 [PITH_FULL_IMAGE:figures/full_fig_p022… view at source ↗
Figure 6
Figure 6. Figure 6: Comparison of the reference and predicted solutions of the nonlinear Schrödinger equation for the modulus field |𝜓(𝑥, 𝑡)| at three different time instances: 𝑡 = 0.59 s, 𝑡 = 0.79 s, and 𝑡 = 0.98 s. The proposed SCV-PINN predictions closely agree with the reference solution at all temporal snapshots. 5.2. Forward Problem: Nonlinear Poisson Equation To further examine the capability of the proposed split comp… view at source ↗
Figure 7
Figure 7. Figure 7: Exact solution, SCV-PINN predicted solution, and corresponding absolute error for the nonlinear Poisson equation on a regular domain with 𝑘 = 4. The proposed SCV-PINN achieves a relative 𝐿2 error of 4.45 × 10−7 . 5.3. Forward Problem: Poisson Equation on an L-Shaped Domain To further assess the capability of the proposed split complex-valued physics-informed neural network (SCV￾PINN) in irregular geometrie… view at source ↗
Figure 8
Figure 8. Figure 8: Comparison of the reference solution, SCV-PINN predicted solution, and corresponding absolute error for the Poisson equation on an irregular L-shaped domain. 5.4. Forward Problem: Viscous Burgers’ Equation To evaluate the performance of the proposed split complex-valued physics-informed neural network (SCV-PINN) for nonlinear convection–diffusion problems, we consider the one-dimensional viscous Burgers’ e… view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of the reference solution, predicted solution, and corresponding absolute error for the 1-D Burgers’ equation. The proposed SCV-PINN accurately captures the nonlinear shock behavior with a relative 𝐿2 error of 6.73 × 10−5 [PITH_FULL_IMAGE:figures/full_fig_p026_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Comparison of the reference and predicted solutions of the 1-D Burgers’ equation at four different time instants: 𝑡 = 0.000 s, 𝑡 = 0.333 s, 𝑡 = 0.667 s, and 𝑡 = 1.000 s. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 26 of 54 [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Comparison of the reference solution, predicted solution, and corresponding absolute error for the Allen–Cahn equation. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 27 of 54 [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of the Allen–Cahn solution at three different time instants: 𝑡 = 0.10 s, 𝑡 = 0.45 s, and 𝑡 = 0.90 s. 5.6. Forward Problem: Korteweg–de Vries (KdV) Equation To assess the capability of the proposed split complex-valued physics-informed neural network (SCV-PINN) in modeling nonlinear dispersive wave phenomena, we consider the Korteweg–de Vries (KdV) equation 𝛼 𝜕𝑢 𝜕𝑡 + 𝛽𝑢 𝜕𝑢 𝜕𝑥 + 𝛾 𝜕 3𝑢 𝜕𝑥3 = 0, (… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of the reference solution, predicted solution, and corresponding absolute error for the Korteweg– de Vries (KdV) equation. 5.7. Forward Problem: Helmholtz Equation To further investigate the capability of the proposed split complex-valued physics-informed neural network (SCV￾PINN) framework in modeling wave propagation phenomena, we consider the two-dimensional Helmholtz equation defined over t… view at source ↗
Figure 14
Figure 14. Figure 14: Comparison of the exact and predicted solutions of the Helmholtz equation (regular) for 𝑎1 = 1 and 𝑎2 = 4 [PITH_FULL_IMAGE:figures/full_fig_p030_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Comparison of the exact and predicted slices of the Helmholtz equation (regular) at 𝑦 = 0.5, 𝑦 = 0.0, and 𝑦 = −0.5. 5.8. Forward Problem: Helmholtz Equation over a Square Domain with a Circular Hole To further investigate the geometric flexibility and boundary handling capability of the proposed split complex￾valued physics-informed neural network (SCV-PINN) framework, we consider the Helmholtz equation d… view at source ↗
Figure 16
Figure 16. Figure 16: Comparison of the exact solution, predicted solution, and corresponding absolute error for the Helmholtz equation over a domain with a circular hole of radius 𝑅 = 1∕4. 5.9. Forward Problem: Kovasznay Flow at Re = 20 To assess the performance of the proposed split complex-valued physics-informed neural network (SCV-PINN) for incompressible fluid flow problems, we consider the steady two-dimensional Kovaszn… view at source ↗
Figure 17
Figure 17. Figure 17: Comparison of the exact and predicted 𝑢-velocity, 𝑣-velocity, and pressure fields for the Kovasznay flow at 𝑅𝑒 = 20, along with the corresponding absolute error. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 33 of 54 [PITH_FULL_IMAGE:figures/full_fig_p033_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Comparison of the exact and predicted streamline of the Kovasznay flow at 𝑅𝑒 = 20. 5.10. Forward Problem: Lid-Driven Cavity Flow at Re = 100 To further assess the capability of the proposed split complex-valued physics-informed neural network (SCV￾PINN) for incompressible viscous flows, we consider the classical lid-driven cavity benchmark at Re = 100. Owing to the presence of strong velocity gradients, v… view at source ↗
Figure 19
Figure 19. Figure 19: Comparison of the benchmark and predicted 𝑢-velocity, 𝑣-velocity, and pressure fields for the lid-driven cavity flow at 𝑅𝑒 = 100. The proposed SCV-PINN accurately captures the incompressible flow structures and pressure distribution. The corresponding relative 𝐿2 errors for 𝑢, 𝑣, and 𝑝 are 2.64 × 10−3 , 8.53 × 10−3, and 6.04 × 10−2 , respectively. 5.11. Inverse Problem: Lorenz Dynamical System To assess t… view at source ↗
Figure 20
Figure 20. Figure 20: Inverse identification history of the Lorenz system parameters: Prandtl number (𝜆1 ), Rayleigh number (𝜆2 ), and geometric aspect ratio (𝜆3 ) with respect to training epochs. The exact values are (10.000, 15.000, 2.667), while the proposed SCV-PINN predicts (10.001, 15.000, 2.667), showing excellent agreement with the reference parameters. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 37 of… view at source ↗
Figure 21
Figure 21. Figure 21: Lorenz system dynamics: left, numerical data points; right, predicted solution trajectory obtained using the proposed SCV-PINN. The proposed framework accurately captures the chaotic butterfly-shaped attractor and nonlinear temporal evolution. 5.12. Inverse Problem: Burgers’ Equation As a second inverse benchmark, we consider the one-dimensional Burgers’ equation 𝑢𝑡 + 𝜆1 𝑢𝑢𝑥 − 𝜆2 𝑢𝑥𝑥 = 0, (164) defined on… view at source ↗
Figure 22
Figure 22. Figure 22: Inverse Burgers’ equation results obtained using the proposed SCV-PINN with a 7-hidden-layer and 20- neurons-per-layer architecture. The top row shows the exact solution with 2000 observation data points (left), predicted solution (middle), and corresponding absolute error (right). The bottom row presents the reference and predicted solution comparisons at 𝑡 = 0.25 s, 𝑡 = 0.50 s, and 𝑡 = 0.75 s [PITH_FUL… view at source ↗
Figure 23
Figure 23. Figure 23: Convergence history of the Burgers’ inverse problem parameters: convection coefficient (𝜆1 ) and diffusion coefficient (𝜆2 ) with respect to training epochs. The comparison between PINN and the proposed SCV-PINN is performed using the same architecture consisting of 8 hidden layers with 20 neurons per layer. 5.13. Inverse Problem: Navier–Stokes Equation To assess the inverse modeling capability of the pro… view at source ↗
Figure 24
Figure 24. Figure 24: Convergence history of the identified parameters (𝜆1 , 𝜆2 ) with respect to training epochs for the two￾dimensional Navier–Stokes inverse problem. The left and right panels correspond to 𝜆1 and 𝜆2 , respectively. The exact parameter values are 𝜆1 = 1.00137 and 𝜆2 = 0.01007. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 41 of 54 [PITH_FULL_IMAGE:figures/full_fig_p041_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Inverse Navier–Stokes flow field reconstruction at 𝑅𝑒 = 100: comparison of the benchmark and predicted 𝑢-velocity, 𝑣-velocity, and pressure fields together with their corresponding absolute error distributions obtained using the proposed SCV-PINN. 5.14. 3D Navier–Stokes Equation: Beltrami Flow To further assess the robustness and versatility of the proposed split complex-valued physics-informed neural net… view at source ↗
Figure 26
Figure 26. Figure 26: Comparison of the exact and predicted 𝑢-velocity, 𝑣-velocity, 𝑤-velocity, and pressure fields together with their corresponding absolute error distributions for the three-dimensional Navier–Stokes Beltrami flow obtained using the proposed SCV-PINN. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 44 of 54 [PITH_FULL_IMAGE:figures/full_fig_p044_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Training loss convergence histories of the conventional PINN and the proposed SCV-PINN frameworks. The proposed SCV-PINN consistently exhibits faster convergence and lower final loss values across a range of benchmark problems. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 46 of 54 [PITH_FULL_IMAGE:figures/full_fig_p046_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Training loss convergence histories obtained using the conventional PINN and the proposed SCV-PINN frameworks for (a) the Burgers’ forward problem and (b) the Burgers’ inverse problem [PITH_FULL_IMAGE:figures/full_fig_p047_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Comparison of the relative 𝐿2 error obtained using the conventional PINN, SA-PINN, and the proposed SCV￾PINN frameworks under identical network architectures and training hyperparameters for the Burgers’ equation (forward problem). Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 47 of 54 [PITH_FULL_IMAGE:figures/full_fig_p047_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Comparison of different split complex-valued activation functions and collocation point sampling strategies employed in the proposed SCV-PINN framework. The figure illustrates the influence of activation selection and sampling distributions on the training convergence and solution accuracy. Barman, Ray, Chatterjee: Preprint submitted to Elsevier Page 50 of 54 [PITH_FULL_IMAGE:figures/full_fig_p050_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Distribution of collocation and initial–boundary training points for the Burgers’ forward problem using different non-adaptive sampling strategies, including random, grid, Latin hypercube sampling (LHS), Sobol, Halton, and Hammersley sequences. The figure illustrates the spatial coverage and point distribution characteristics of each sampling method within the SCV-PINN framework. Barman, Ray, Chatterjee: … view at source ↗

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This paper was first reviewed by deepseek-v4-flash on August 2, 2026.