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arxiv: 2604.25606 · v1 · submitted 2026-04-28 · 🧮 math.NA · cs.NA

C-PINN: A neural network framework based on the Cord\`{e}s condition for solving linear and fully nonlinear equations in non-divergence form and its applications

Bingcheng Hu , Lixiang Jin , Zhaoxiang Li This is my paper

Pith reviewed 2026-05-07 15:35 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords physics-informed neural networksCordès conditionnon-divergence form PDEsMonge-Ampère equationsHamilton-Jacobi-Bellman equationsoptimal transportnumerical PDE methods
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The pith

Embedding the Cordès condition into the loss function lets physics-informed neural networks solve non-divergence form PDEs more stably and accurately.

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

The paper introduces C-PINN, a framework that adapts physics-informed neural networks for both linear and fully nonlinear partial differential equations written in non-divergence form. It works by placing information about the structure of the differential operator, drawn from the Cordès condition, directly into the loss used during training. The goal is to improve how well-conditioned the optimization problem is, which in turn makes training more stable and produces more accurate solutions. The same idea is applied to Hamilton-Jacobi-Bellman and Monge-Ampère equations and tested on optimal-transport problems, with results shown for high-dimensional cases.

Core claim

By incorporating the operator structure into the loss function via the Cordès condition, the proposed neural network framework improves the conditioning of the associated optimization problem, thereby enhancing training stability and solution accuracy for linear and fully nonlinear PDEs in non-divergence form.

What carries the argument

The Cordès condition, a structural property of the PDE operator that is folded into the PINN loss function to reflect the non-divergence form directly.

If this is right

  • The framework extends directly to Hamilton-Jacobi-Bellman equations and Monge-Ampère equations.
  • It supports applications such as optimal transport problems.
  • Numerical results indicate the method remains effective for high-dimensional instances.
  • Its generality and simplicity allow use across a range of scientific and engineering PDE problems.

Where Pith is reading between the lines

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

  • The same conditioning idea could be tried on other classes of nonlinear PDEs that standard PINNs handle poorly.
  • Combining C-PINN with adaptive collocation or residual-based sampling might further reduce training cost.
  • If the improvement scales, it could reduce reliance on problem-specific network architectures for certain non-divergence equations.

Load-bearing premise

That embedding the Cordès condition into the PINN loss function will reliably improve conditioning and accuracy for both linear and fully nonlinear non-divergence PDEs without introducing new instabilities or requiring problem-specific tuning.

What would settle it

A controlled numerical test on a non-divergence PDE where standard PINN training succeeds but the C-PINN version diverges or yields higher error would falsify the claimed improvement in conditioning and accuracy.

Figures

Figures reproduced from arXiv: 2604.25606 by Bingcheng Hu, Lixiang Jin, Zhaoxiang Li.

Figure 3.1
Figure 3.1. Figure 3.1: The loss landscape of the HJB equation task With the theoretical framework and the modified objective function of C-PINN now fully es￾tablished, we proceed to evaluate its effectiveness, accuracy, and robustness through a series of comprehensive numerical experiments in the following section. A comprehensive visual summary of the proposed framework is illustrated in view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Dual-Loop C-PINN framework 4. Numerical experiments 4.1. Training Strategy and Optimization Metrics. In this section, we present numerical ex￾periments to verify the effect of the C-PINN by solving various linear non-divergence equations, Hamilton-Jacobi-Bellman equations and Monge-Amp`ere equations. Additionally, we compare the performance of C-PINN and PINN in these tasks. To effectively minimize the p… view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Results for the smooth case of (4.4). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.2
Figure 4.2. Figure 4.2: Optimization dynamics comparison: C-PINN vs. PINN for the smooth case of (4.4) view at source ↗
Figure 4.3
Figure 4.3. Figure 4.3: Results for the weakly singular case of (4.4). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.4
Figure 4.4. Figure 4.4: Optimization dynamics comparison: C-PINN vs. PINN for the weakly singular case of (4.4) view at source ↗
Figure 4
Figure 4. Figure 4: , which demonstrates the significantly enhanced landscape smoothness and gradient stability view at source ↗
Figure 4.5
Figure 4.5. Figure 4.5: Results for problem Eq. (4.7). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.6
Figure 4.6. Figure 4.6: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.7) While the case with continuous coefficients provides a baseline, we further investigate the more challenging scenario with discontinuous coefficients to assess the robustness of the proposed method across different regularity regimes. Let A = view at source ↗
Figure 4.7
Figure 4.7. Figure 4.7: Results for problem Eq. (4.8). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.8
Figure 4.8. Figure 4.8: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.8) 4.4. Example 4.3 High Dimensional Equations. To evaluate the scalability and robustness of the proposed C-PINN framework in higher dimensions and over complex, non-cuboid geometries, we consider a three-dimensional linear non-divergence elliptic equation. Let the computational domain be an ellipsoid defined by Ω = {(x1, x2, x3) ∈ R 3… view at source ↗
Figure 4.9
Figure 4.9. Figure 4.9: Results for problem Eq. (4.10). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.10
Figure 4.10. Figure 4.10: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.10) While the preceding 3D experiments demonstrated C-PINN’s capability to handle complex ge￾ometries and low-regularity features, a critical advantage of neural-network-based solvers is their potential to overcome the curse of dimensionality. To evaluate this scalability, our subsequent experiments consider higher-dimensional equation… view at source ↗
Figure 4.11
Figure 4.11. Figure 4.11: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.12) To push the boundaries of our methodology and explicitly demonstrate its capability to circum￾vent the curse of dimensionality, we scale the problem up to a 20D space. Consider another 20D case as follows: A(x) = (aij )20×20 = ( 20, if i = j xixj |xixj | , if i ̸= j for i, j ∈ {1, 2, . . . , 20}, (4.14) which satisfies Eq. (1.1), a… view at source ↗
Figure 4.12
Figure 4.12. Figure 4.12: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.14) In the previous examples, the exact solution is known, which allows for a direct evaluation of the approximation error. However, in many practical problems, the exact solution is unavailable. To better assess the performance of the proposed method in such realistic scenarios, we consider elliptic equations with a constant source te… view at source ↗
Figure 4.13
Figure 4.13. Figure 4.13: Comparison of numerical solutions for problem (4.17). Left: Results obtained via C-PINN; Right: Results obtained via the Chebyshev spectral method After that, we consider the discontinuous coefficients with unknown solution. In this case, we consider the domain to be Ω = (−1, 1) × (−1, 1), and the coefficient is b = (x1, x2) T , c = 3, and A = view at source ↗
Figure 4.14
Figure 4.14. Figure 4.14: Comparison of numerical solutions for problem (4.18). Left: Results obtained via C-PINN; Right: Results obtained via the Chebyshev spectral method Building upon the previous experiments on linear elliptic equations, we further investigate the performance of the proposed method on HJB equations. As a prototypical class of fully nonlinear view at source ↗
Figure 4.15
Figure 4.15. Figure 4.15: Results for problem Eq. (4.19). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.16
Figure 4.16. Figure 4.16: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.19) Building upon the results for HJB equations, we further investigate the MA equations, which serves as a prototypical example of fully nonlinear PDEs with strong structural constraints. Com￾pared to HJB equations, the MA equations introduce additional challenges due to its highly nonlin￾ear and degenerate nature. This example provid… view at source ↗
Figure 4.17
Figure 4.17. Figure 4.17: Results for problem Eq. (4.23). From left to right: exact solution, numerical solution, and absolute error view at source ↗
Figure 4.18
Figure 4.18. Figure 4.18: Optimization dynamics comparison: C-PINN vs. PINN for problem Eq. (4.23) Building upon the numerical results for the Monge-Amp`ere equation with Dirichlet boundary conditions, we further explore its role in optimal transport problems. As is well known, the Monge￾Amp`ere equation provides a fundamental link between convex potential functions and optimal trans￾port maps. By considering this application, w… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Source density and transported grid under the optimal transport map Under the action of the optimal transport map, the grid undergoes a significant redistribution: the initially uniform Cartesian grid becomes denser in low-density regions and sparser in high￾density regions. This behavior provides a clear illustration of the mass conservation mechanism, whereby the transport map adjusts local volume elem… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: 3D surface models of the cortical surface and the lion head view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Comparison of conformal and optimal transport mappings 6. Conclusion In this work, we propose an improved PINN framework based on the Cord`es condition, demon￾strating its effectiveness in solving both linear and fully nonlinear PDEs. The proposed C-PINN enhances the loss formulation and provides a stable and accurate numerical framework for problems in non-divergence form and beyond. The key contributio… view at source ↗
read the original abstract

In this paper, we propose a novel Physics-Informed Neural Network (PINN) framework based on the Cord\`{e}s condition for solving both linear and fully nonlinear partial differential equations (PDEs) in non-divergence form, together with their applications. By incorporating the operator structure into the loss function, the proposed method improves the conditioning of the associated optimization problem, thereby enhancing training stability and solution accuracy. The framework is further extended to include Hamilton-Jacobi-Bellman and Monge-Amp\`{e}re equations, with applications to optimal transport. Numerical experiments demonstrate the effectiveness and robustness of the method, as well as its capability to address high-dimensional problems, highlighting the promise of learning-based approaches for tackling challenging PDEs. Owing to its generality and simplicity, the proposed method is expected to be of broad interest to the scientific and engineering communities.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes C-PINN, a physics-informed neural network framework that embeds the Cordès condition into the loss function to solve linear and fully nonlinear PDEs in non-divergence form. It claims this incorporation improves the conditioning of the associated non-convex optimization problem, thereby enhancing training stability and solution accuracy. The framework is extended to Hamilton-Jacobi-Bellman and Monge-Ampère equations with applications to optimal transport, and numerical experiments are presented to demonstrate effectiveness, robustness, and applicability to high-dimensional problems.

Significance. If the claimed conditioning improvement and resulting accuracy gains hold under rigorous verification, the work could offer a practical extension of PINNs to a class of PDEs that are difficult for standard divergence-form methods, with potential value for high-dimensional optimal transport and fully nonlinear problems. The generality across linear and nonlinear cases is a strength, though the absence of direct diagnostics for the core innovation limits the current impact.

major comments (2)
  1. [Numerical experiments] Numerical experiments section: The manuscript reports accuracy results on linear and nonlinear test cases but provides no quantitative diagnostics (such as loss Hessian condition numbers, eigenvalue spreads, or side-by-side optimization trajectory comparisons) that isolate the effect of the Cordès term versus standard PINN losses, network architecture, or hyperparameter choices. This directly undermines support for the central claim that embedding the Cordès condition improves conditioning and stability.
  2. [Loss function construction] Loss function construction (likely §3): The presentation does not clarify whether the Cordès incorporation introduces additional problem-specific scaling or weighting parameters that must be tuned per PDE, which risks contradicting the claimed generality and simplicity without introducing new instabilities.
minor comments (2)
  1. [Abstract] Abstract: The statement that 'numerical experiments demonstrate the effectiveness and robustness' is unsupported by any specific error metrics, baselines, or quantitative highlights, reducing clarity for readers.
  2. Notation: Ensure consistent use of the Cordès condition symbol and its embedding in the loss across equations and text to avoid ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us identify areas for improvement. We address each major comment below.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: The manuscript reports accuracy results on linear and nonlinear test cases but provides no quantitative diagnostics (such as loss Hessian condition numbers, eigenvalue spreads, or side-by-side optimization trajectory comparisons) that isolate the effect of the Cordès term versus standard PINN losses, network architecture, or hyperparameter choices. This directly undermines support for the central claim that embedding the Cordès condition improves conditioning and stability.

    Authors: We acknowledge that direct quantitative diagnostics, such as Hessian condition numbers or eigenvalue spreads, would provide stronger isolation of the Cordès term's effect on optimization conditioning. Our current experiments demonstrate consistent gains in accuracy and training stability across linear and nonlinear cases, but these are indirect. In the revised manuscript we will add side-by-side training-loss curves and, for selected low-dimensional examples, approximate condition-number estimates of the loss Hessian to furnish more direct evidence. revision: yes

  2. Referee: [Loss function construction] Loss function construction (likely §3): The presentation does not clarify whether the Cordès incorporation introduces additional problem-specific scaling or weighting parameters that must be tuned per PDE, which risks contradicting the claimed generality and simplicity without introducing new instabilities.

    Authors: The Cordès term is incorporated by augmenting the standard PDE residual with an operator-derived expression that follows directly from the Cordès condition; no additional problem-specific scaling or weighting coefficients are introduced beyond the conventional balancing weights already used in PINN losses. We will revise Section 3 to display the explicit loss expression and state that the hyperparameter choices remain identical in form to those of standard PINNs, thereby preserving the claimed generality. revision: partial

Circularity Check

0 steps flagged

No significant circularity; novel framework construction independent of inputs

full rationale

The paper presents C-PINN as a new construction that augments the standard PINN loss with the Cordès condition to encode operator structure. No step reduces a claimed prediction or uniqueness result to a fitted quantity defined by the target output, nor does any load-bearing premise collapse to a self-citation chain. The central claim (conditioning improvement via the modified loss) is an empirical hypothesis tested on linear/nonlinear examples rather than an identity by construction. Self-citations, if present, are not required to justify the framework itself. This is the common case of an honest new method whose verification gaps (e.g., missing Hessian diagnostics) concern correctness, not circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No details on free parameters, axioms, or invented entities are provided in the abstract.

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