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arxiv: 2604.27731 · v1 · submitted 2026-04-30 · 🧮 math.NA · cs.NA

An adaptive Deep Ritz framework for second-order fully nonlinear partial differential equations

Alexandre Caboussat , Martin T. Leclercq , Anna Peruso This is my paper

Pith reviewed 2026-05-07 06:47 UTC · model grok-4.3

classification 🧮 math.NA cs.NA
keywords Deep Ritz methodfully nonlinear PDEsMonge-Ampère equationadaptive samplingleast-squares splittingneural networks for PDEsoptimal transport
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The pith

Splitting algorithm decouples nonlinear PDEs for Deep Ritz neural network solutions

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

The paper proposes an adaptive Deep Ritz framework to solve second-order fully nonlinear partial differential equations as an alternative to PINNs. It uses a least-squares algorithm to split the problem into local nonlinear solves handled by standard methods and linear variational subproblems addressed by a Deep Ritz neural network. An adaptive sampling strategy selects collocation points to improve efficiency. The approach is tested on the Dirichlet problem for the Monge-Ampère equation and extended to optimal transport problems with transport boundary conditions, with direct comparisons to full PINNs implementations.

Core claim

A least-squares splitting method decouples the nonlinearities from the variational features of fully nonlinear PDEs, enabling iterative solution of local nonlinear problems alongside linear variational problems solved via a Deep Ritz neural network, with adaptive sampling of collocation points to maintain accuracy while increasing efficiency; this is demonstrated for the Monge-Ampère Dirichlet problem and the optimal transport variant.

What carries the argument

The least-squares splitting algorithm that separates local nonlinear problems from linear variational subproblems solved by Deep Ritz neural networks with adaptive collocation point selection.

If this is right

  • The framework applies to multiple fully nonlinear equations by reusing existing nonlinear solvers for the local steps.
  • Adaptive sampling reduces the number of collocation points needed without sacrificing solution accuracy.
  • Direct comparisons show the variational Deep Ritz component can outperform or complement full PINN training on the same problems.
  • The method extends naturally to optimal transport formulations with adjusted boundary conditions.

Where Pith is reading between the lines

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

  • Hybrid traditional-numerical plus neural solvers may scale better for problems where nonlinearities dominate in isolated regions.
  • The splitting idea could be tested on other variational PDE solvers to reduce overall training cost.
  • Adaptive point selection might combine with error estimators from finite element theory for further gains.

Load-bearing premise

That the splitting preserves the variational structure so the Deep Ritz network can accurately solve the resulting linear subproblems without introducing significant errors.

What would settle it

If the iterative method fails to converge to a known exact solution for the Monge-Ampère equation under standard Dirichlet conditions while a full PINN approach succeeds, the decoupling benefit would be invalidated.

Figures

Figures reproduced from arXiv: 2604.27731 by Alexandre Caboussat, Anna Peruso, Martin T. Leclercq.

Figure 1
Figure 1. Figure 1: Illustration of the Monge optimal transport problem (in 1D), which consists in finding the optimal transport map T that transports the mass µ0 onto µ1. When the cost is the quadratic cost, Brenier’s theorem, see, e.g., [31], gives a convenient PDE formu￾lation of the optimal transport problem under standard regularity and support assumptions. If µ0 and µ1 are absolutely continuous probability measures on R… view at source ↗
Figure 2
Figure 2. Figure 2: Schematic of an Input Convex Neural Network (ICNN) and notation. The convexity of the solution is then guaranteed by two elementary facts: i) nonnegative linear combinations of convex functions are convex; ii) the composition of a convex function with an increasing convex function yields a convex function. A specific aspect of ICNNs is the use of passthrough layers {L (l)}l , which provide direct connectio… view at source ↗
Figure 3
Figure 3. Figure 3: First exponential test case, with α = 1 and 3000 collocation points. We illustrate the loss function for each epoch and the errors between u and uNN computed only at the end of each splitting iteration. The same total number of epochs is used for PINNs. The shadowed area represents the 5th-95th quantile. Finally, view at source ↗
Figure 4
Figure 4. Figure 4: First exponential test case, with α = 1. Deep Ritz method. The curves correspond to various values of the seeds (1% seeds corresponds to S = nc/100). In a second step, we consider the case α = 4 to introduce sharper gradients view at source ↗
Figure 5
Figure 5. Figure 5: First exponential test case, with α = 1, 3000 collocation points and 300 boundary points. Pointwise absolute error at the end of each splitting iteration. Top row: no adaptive sampling; Bottom row: with adaptive sampling view at source ↗
Figure 6
Figure 6. Figure 6: First exponential test case, with α = 1, 3000 collocation points and 300 boundary points. Distribution of the resampled points (the seeds are illustrated in black) view at source ↗
Figure 7
Figure 7. Figure 7: First exponential test case, with α = 4, 3000 collocation points and 300 boundary points. We illustrate the loss function for each epoch and the errors between u and uNN computed only at the end of each splitting iteration. The shadowed area represents the 5th-95th quantile view at source ↗
Figure 8
Figure 8. Figure 8: Second test case, with R = 2, 3000 collocation points and 300 boundary points. We illustrate the loss function for each epoch and the errors between u and uNN computed only at the end of each splitting iteration. The shadowed area represents the 5th-95th quantile. (a) Loss function. (b) L 2 error. (c) H2 error view at source ↗
Figure 9
Figure 9. Figure 9: Second test case, with R = 2. Deep Ritz method. The curves corresponds various values of the seeds (1% seeds corresponds to S = nc/100) view at source ↗
Figure 10
Figure 10. Figure 10: Second test case, with R = 2, 3000 collocation points and 300 boundary points. Deep Ritz method. Pointwise absolute error at the end of each splitting iteration. Top row: without adaptive sampling; bottom row: with adaptive sampling. In a second step, let us consider the case with R = √ 2 + 0.01 that is more stringent as it is close to the singular case R = √ 2 view at source ↗
Figure 11
Figure 11. Figure 11: Second test case, with R = 2, 3000 collocation points and 300 boundary points. Deep Ritz method. Distribution of the resampled points (the seeds are illustrated in black). (a) PINNs. 3000 points. (b) DR. 3000 points view at source ↗
Figure 12
Figure 12. Figure 12: Second test case, with R = √ 2 + 0.01, 3000 collocation points and 300 boundary points. We illustrate the loss function for each epoch and the errors between u and uNN computed only at the end of each splitting iteration. The shadowed area represents the 5th-95th quantile view at source ↗
Figure 13
Figure 13. Figure 13: A test case on the unit disk, with 3000 collocation points and 300 boundary points, together with adaptive sampling. We illustrate the loss function for each epoch and the errors between u and uNN computed only at the end of each splitting iteration. The shadowed area represents the 5th-95th quantile view at source ↗
Figure 14
Figure 14. Figure 14: A test case on the unit disk, with 3000 collocation points and 300 boundary points. Deep Ritz method. Pointwise absolute error at the end of each splitting iteration. Top row: without adaptive sampling; bottom row: with adaptive sampling view at source ↗
Figure 15
Figure 15. Figure 15: Test case for the Pucci’s equation. Convergence results when using the Deep Ritz method with 3000 collocation points and 300 boundary points. 6.5. Extension to the Gauss curvature equation in 2D: The Minkowski problem. Let us consider Ω = [0, 1]2 , and the data b = view at source ↗
Figure 16
Figure 16. Figure 16: Gauss curvature equation in 2D. Convergence results with the adaptive sam￾pling procedure, 3000 collocation points and 300 boundary points. (a) Without adaptive sampling. (b) With adaptive sampling view at source ↗
Figure 17
Figure 17. Figure 17: Gauss curvature equation in 2D, with 3000 collocation points and 300 bound￾ary points. Deep Ritz method. Pointwise absolute error at the end of each splitting iter￾ation. Top row: without adaptive sampling; bottom row: with adaptive sampling view at source ↗
Figure 18
Figure 18. Figure 18: Optimal transport Monge-Amp`ere problem. Disk domain into an ellipse. Visualization of the transport map with histograms based on 106 sampling points. Results after 30 iterations of the splitting algorithm. Left: Source distribution f; middle: Exact target distribution g; right: approximated target distribution (∇uNN )#(f) view at source ↗
Figure 19
Figure 19. Figure 19: Optimal transport Monge-Amp`ere problem. Disk domain into an ellipse. Visualization of the approximated vector field ∇uNN . First component (∇uNN )x Second component (∇uNN )y view at source ↗
Figure 20
Figure 20. Figure 20: Optimal transport Monge-Amp`ere problem. Disk domain into an ellipse. Vi￾sualisation of the components of the approximated vector field ∇uNN . Left: components of ∇uNN ; right: approximation error for each component. 7.2. Optimal transport Monge-Amp`ere: transporting a Gaussian distribution into a uniform distribution. In an effort to mimic the optimal transport of piles of debris [30], we transport a Gau… view at source ↗
Figure 21
Figure 21. Figure 21: Optimal transport Monge-Amp`ere problem. Gaussian distribution into uni￾form. Visualization of the transport of the density function at several iterations of the splitting algorithm with histograms based on 106 sampling points. Top row: without adaptive sampling; bottom row: with adaptive sampling. First component (∇uNN )x Second component (∇uNN )y view at source ↗
Figure 22
Figure 22. Figure 22: Optimal transport Monge-Amp`ere problem. Gaussian distribution into uni￾form. Without adaptive sampling. Visualisation of the components of the approximated vector field ∇uNN . Left: components of ∇uNN ; right: approximation error for each com￾ponent after 20 splitting iterations. 7.3. Optimal transport Monge-Amp`ere: transporting two Gaussian distributions into a uni￾form distribution. Second, in an effo… view at source ↗
Figure 23
Figure 23. Figure 23: Optimal transport Monge-Amp`ere problem. Gaussian distribution into uni￾form. With adaptive sampling. Visualisation of the components of the approximated vector field ∇uNN . Left: components of ∇uNN ; right: approximation error for each com￾ponent after 20 splitting iterations. density f initially (left), and the transport of those points at different iterations of the algorithm, with and without the adap… view at source ↗
Figure 24
Figure 24. Figure 24: Optimal transport Monge-Amp`ere problem. Two Gaussian distributions into uniform. Visualization of the transport of the density function at several iterations of the splitting algorithm with histograms based on 106 sampling points. Top row: without adaptive sampling; bottom row: with adaptive sampling. 7.4. Optimal transport Monge-Amp`ere: transporting two Gaussian distributions into a Gauss￾ian distribut… view at source ↗
Figure 25
Figure 25. Figure 25: Optimal transport Monge-Amp`ere problem. Two Gaussian distributions into one Gaussian distribution. Visualization of the transport of the density function at several iterations of the splitting algorithm with histograms based on 106 sampling points. Top row: without adaptive sampling; bottom row: with adaptive sampling. 8. Conclusions We have presented a novel algorithm based on a least-squares approach t… view at source ↗
read the original abstract

As an alternative to PINNs, a Deep Ritz framework is proposed to solve fully nonlinear PDEs. A least-squares algorithm is advocated to decouple the nonlinearities from the variational features of several fully nonlinear PDEs. A splitting method allows to iteratively solve local nonlinear problems and linear variational problems at each iteration. While existing nonlinear solvers are applied to solve for nonlinearities, we propose a novel coupling with a Deep Ritz neural network approach that is well-suited to the variational flavor of the linear variational problems. An adaptive sampling strategy for the selection of collocation points is incorporated to increase the efficiency of the algorithm without sacrificing its accuracy. Numerical experiments are presented to solve the Dirichlet problem for several fully nonlinear equations, starting with the prototypical Monge-Amp\`ere equation, showing the flexibility of the approach. Numerical results are compared with results obtained using a full PINNs approach. Finally, numerical experiments are extended to address the optimal transport Monge-Amp\`ere problem with transport boundary conditions.

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 / 3 minor

Summary. The manuscript proposes an adaptive Deep Ritz framework as an alternative to PINNs for second-order fully nonlinear PDEs. A least-squares formulation decouples the nonlinearities, enabling a splitting iteration that alternates between local nonlinear solves (via standard solvers) and linear variational subproblems (discretized by a Deep Ritz neural network). An adaptive collocation strategy selects points for the variational problems. Numerical experiments solve Dirichlet problems for the Monge-Ampère equation and extend to the optimal transport problem with transport boundary conditions, with direct comparisons to a full PINN approach.

Significance. If validated, the approach could offer a more variationally natural and potentially efficient alternative to PINNs for fully nonlinear equations by exploiting the Deep Ritz method on the linear subproblems and incorporating adaptivity. The numerical demonstrations on Monge-Ampère and optimal transport problems illustrate flexibility across boundary conditions. However, the complete absence of convergence analysis or error estimates for the coupled splitting-plus-NN iteration substantially limits the work's significance in numerical analysis, where such guarantees are standard for iterative schemes applied to nonlinear PDEs.

major comments (2)
  1. [Section 2] Section 2 (algorithm description): No convergence analysis, contraction argument, or error propagation bound is given for the splitting iteration when the linear variational subproblems are replaced by Deep Ritz neural-network approximations. It is therefore unclear whether the iterates converge to a solution of the original fully nonlinear PDE (e.g., Monge-Ampère) in the presence of NN approximation error and adaptive point selection bias. This is load-bearing for the central claim that the framework reliably solves second-order fully nonlinear equations.
  2. [Section 3] Section 3 (numerical experiments): The comparisons with PINNs are presented only through selected plots and qualitative statements; no quantitative error tables, convergence rates with respect to network width/depth or number of collocation points, or ablation studies isolating the effect of the adaptive sampler appear. Without these data it is impossible to substantiate the claim that adaptive sampling increases efficiency without sacrificing accuracy.
minor comments (3)
  1. [Section 2.1] The least-squares functional used for decoupling should be written explicitly with all terms (including any regularization) so that readers can verify it is indeed parameter-free and equivalent to the original PDE.
  2. Figure captions and legends in the numerical section should explicitly state the network architecture, optimizer, and number of adaptive iterations used for each example to improve reproducibility.
  3. A short pseudocode box summarizing the overall splitting loop (nonlinear solve + Deep Ritz solve + adaptive resampling) would clarify the coupling between the components.

Simulated Author's Rebuttal

2 responses · 1 unresolved

We thank the referee for their thorough review and valuable feedback on our manuscript. We appreciate the recognition of the potential of the adaptive Deep Ritz framework as an alternative to PINNs for fully nonlinear PDEs. We address the major comments below and will make revisions to improve the manuscript, particularly by enhancing the quantitative aspects of the numerical experiments and adding discussion on the convergence properties.

read point-by-point responses

  1. Referee: Section 2 (algorithm description): No convergence analysis, contraction argument, or error propagation bound is given for the splitting iteration when the linear variational subproblems are replaced by Deep Ritz neural-network approximations. It is therefore unclear whether the iterates converge to a solution of the original fully nonlinear PDE (e.g., Monge-Ampère) in the presence of NN approximation error and adaptive point selection bias. This is load-bearing for the central claim that the framework reliably solves second-order fully nonlinear equations.

    Authors: We agree that the lack of a rigorous convergence analysis for the splitting iteration in the presence of neural network approximations and adaptive sampling is a significant point. The manuscript is primarily focused on developing and demonstrating the algorithmic framework numerically. In the revised version, we will add a discussion in Section 2 on the theoretical foundations, including references to convergence results for the Deep Ritz method and for splitting schemes in nonlinear problems. We will also provide heuristic arguments on why the combined errors are controlled in practice, supported by the numerical results. A full mathematical proof of convergence for the coupled system is beyond the current scope and would constitute a separate theoretical paper; we note this limitation explicitly in the revision. revision: partial

  2. Referee: Section 3 (numerical experiments): The comparisons with PINNs are presented only through selected plots and qualitative statements; no quantitative error tables, convergence rates with respect to network width/depth or number of collocation points, or ablation studies isolating the effect of the adaptive sampler appear. Without these data it is impossible to substantiate the claim that adaptive sampling increases efficiency without sacrificing accuracy.

    Authors: We acknowledge that the numerical comparisons could be strengthened with more quantitative information. We will revise Section 3 to include tables with quantitative error metrics (such as relative L^2 errors and maximum pointwise errors) for the Monge-Ampère equation and the optimal transport problem, comparing our adaptive Deep Ritz method to the PINN approach. Additionally, we will present convergence studies with respect to the number of collocation points and network architecture parameters. An ablation study isolating the adaptive sampling strategy versus uniform sampling will be added to demonstrate its benefits in terms of efficiency and accuracy. revision: yes

standing simulated objections not resolved
  • Full rigorous convergence analysis of the splitting iteration incorporating Deep Ritz approximations and adaptive collocation

Circularity Check

0 steps flagged

No significant circularity; algorithmic extension of standard variational and Deep Ritz methods

full rationale

The paper describes a numerical framework that applies least-squares decoupling and iterative splitting to separate nonlinear local solves from linear variational subproblems, which are then discretized via an existing Deep Ritz neural network plus adaptive collocation. No derivation chain is presented that reduces a claimed prediction or first-principles result to a quantity defined in terms of the method's own fitted parameters or outputs. The approach rests on well-established variational principles for the linear subproblems and prior Deep Ritz literature for the network approximation; numerical experiments on the Monge-Ampère equation and optimal transport variants serve as independent validation rather than self-referential confirmation. No self-definitional steps, fitted-input-as-prediction reductions, or load-bearing self-citations that collapse the central construction are identifiable from the abstract or method outline.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework assumes that standard neural-network approximation properties carry over to the linear variational subproblems after splitting and that adaptive point selection preserves accuracy; these are domain assumptions rather than new axioms or invented entities.

axioms (2)
  • domain assumption Deep Ritz neural networks can accurately solve the linear variational problems that remain after least-squares splitting of the nonlinear PDE.
    Invoked when the paper states that the Deep Ritz approach is well-suited to the variational flavor of the linear subproblems.
  • domain assumption Adaptive selection of collocation points increases efficiency without loss of accuracy for the target class of fully nonlinear equations.
    Stated directly in the abstract as part of the proposed algorithm.

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discussion (0)

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Reference graph

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