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arxiv: 2606.18175 · v1 · pith:2HB4APD4new · submitted 2026-06-16 · 🧮 math.NA · cs.LG· cs.NA· physics.comp-ph

A Convex Quasilinearization Method for Solving Nonlinear PDEs with Physics-Informed Neural Networks

Gbenga T. Awojinrin , Abdul-Akeem Olawoyin , Rami M. Younis This is my paper

Pith reviewed 2026-06-26 23:42 UTC · model grok-4.3

classification 🧮 math.NA cs.LGcs.NAphysics.comp-ph
keywords quasilinearizationphysics-informed neural networksnonlinear PDEsleast-squares collocationNewton-Kantorovich convergencerandom feature modelsconvex optimization
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The pith

Quasilinearization converts nonlinear PDEs into sequences of linear least-squares problems solved directly on linear-in-parameters trial spaces.

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

This paper introduces the LiL-Q method for solving nonlinear PDEs. It uses Bellman-Kalaba quasilinearization to break the nonlinear problem into a sequence of linear subproblems. Each linear subproblem is discretized by collocation on a trial space where the parameters appear linearly, such as random-feature networks or polynomial bases, and solved exactly with a QR factorization. This replaces the nonconvex gradient descent of standard PINNs with convex linear algebra. A convergence analysis shows local convergence under a smallness condition on the initial residual, with the final accuracy determined by how well the trial space approximates the true solution rather than by solver tolerances. The method is tested on seven benchmark problems including Burgers equation, Navier-Stokes, and Darcy flow, typically converging in a handful of outer iterations.

Core claim

The central discovery is that Bellman-Kalaba quasilinearization combined with Linear-in-Learnables trial spaces allows the forward solution of nonlinear PDEs through a short sequence of convex linear least-squares solves whose accuracy is limited solely by the best-approximation residual of the chosen trial space.

What carries the argument

Linear-in-Learnables (LiL) trial spaces, which are representations whose trainable parameters enter linearly into the model, solved by direct QR factorization after quasilinearization reduces the nonlinear PDE to linear subproblems.

If this is right

  • When the exact solution lies exactly in the trial space, the method recovers it to machine precision in a single solve.
  • The outer iteration converges in single-digit steps for most benchmarks, independent of the number of parameters.
  • On the incompressible Navier-Stokes equations the approach matches or exceeds published PINN results while using up to two orders of magnitude fewer trainable parameters.
  • The limiting accuracy is set by the approximation power of the trial space, not by the optimization procedure.

Where Pith is reading between the lines

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

  • The approach may allow stable solutions for stiff nonlinear problems where gradient-based methods struggle with local minima.
  • Extending the LiL spaces to include adaptive or hierarchical bases could further improve efficiency for high-dimensional problems.
  • Since each step is a convex solve, the method could be combined with uncertainty quantification techniques that rely on linear algebra.

Load-bearing premise

The quasilinearization iteration is guaranteed to converge only when the initial guess satisfies an explicit smallness condition on the residual norm.

What would settle it

Demonstrating failure to converge for an initial guess that violates the smallness condition in the Newton-Kantorovich theorem, even when the solution is well-approximated by the trial space, would falsify the local convergence guarantee.

Figures

Figures reproduced from arXiv: 2606.18175 by Abdul-Akeem Olawoyin, Gbenga T. Awojinrin, Rami M. Younis.

Figure 1
Figure 1. Figure 1: Bratu equation: simulated solution fields using P = 25 (top row) and P = 225 (bottom row) for NiL-N, NiL-Q, LiL-N and LiL-Q from left to right. 4.2. 2D Bratu The Bratu equation, introduced as the worked example in Section 3.4, is a nonlinear elliptic PDE that serves as a standard steady model problem. It admits two parameter-dependent solution branches that coalesce for the critical coefficient value λc ≈ … view at source ↗
Figure 2
Figure 2. Figure 2: Bratu equation: convergence histories for various methods using networks with P ∈ {25, 100, 225} parameters. heuristic interventions). LiL-Q converges in 3 iterations at both enriched basis sizes, while LiL-N converges but with a rapidly growing iteration count (1,597 and 7,748 respectively). This suggests that the physical nonlinearity preserves optimization difficulty, even when the architectural nonline… view at source ↗
Figure 3
Figure 3. Figure 3: Bratu equation: convergence histories organized by method [PITH_FULL_IMAGE:figures/full_fig_p020_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Bratu equation: comparison of linearized and nonlinear residual norms for LiL-Q at P ∈ {25, 100, 225}. The asymptotic agreement between the two quantities corroborates the convergence analysis of Section 3.5. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Burgers equation: convergence histories organized by method. Each panel shows all five basis sizes for a single formulation [PITH_FULL_IMAGE:figures/full_fig_p022_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Burgers equation: comparison of linearized and nonlinear residual norms for LiL-Q at P ∈ {25, 100, 225, 400, 625}. 22 [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Burgers equation: predicted solution fields at P = 625 [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Convergence histories for eight LiL configurations on the viscous Burgers equation (P = 625) [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Buckley-Leverett (viscous flow, Ng = 0): convergence histories organized by method. Each panel shows all four basis sizes for a single formulation [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Buckley-Leverett (viscous flow, Ng = 0): comparison of linearized and nonlinear residual norms for LiL-Q at P ∈ {64, 256, 576, 1,024}. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Buckley-Leverett (gravity-influenced flow, Ng = −5): convergence histories organized by method. The NiL-Q panel exhibits order-of-magnitude loss spikes not observed in any other experiment [PITH_FULL_IMAGE:figures/full_fig_p027_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Buckley-Leverett (gravity-influenced flow, Ng = −5): comparison of linearized and nonlinear residual norms for LiL-Q at P ∈ {64, 256, 576, 1,024}. 27 [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Buckley-Leverett solutions at P = 1,024: NiL-N (left) and LiL-Q (right) for viscous displacement (top) and gravity￾influenced displacement (bottom). 28 [PITH_FULL_IMAGE:figures/full_fig_p028_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Linear elasticity at P = 50: exact displacement fields (left), LiL predictions (center), and pointwise absolute errors (right). Errors are at O(10−15) throughout the domain, consistent with machine-precision recovery. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Kovasznay flow at P = 1,875: exact fields (left), LiL-Q predictions (center), and pointwise absolute errors (right) for u, v, and p. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Beltrami flow at t = 1: exact fields (left), LiL-Q predictions (middle), and pointwise absolute errors (right) for u, v, w, and p. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Permeability fields (log10 K in mD) for the four test cases, with heterogeneity contrast increasing from left to right. comparison is indicative rather than definitive and is conservative for LiL-Q, which runs on CPU. Across all four fields, the LiL boundary conditions are satisfied to machine precision by construction, whereas NiL boundary residuals are small but non-zero. The FVM solves each problem in … view at source ↗
Figure 18
Figure 18. Figure 18: Pressure field solutions for (a) FVM (b) NiL and (c) LiL across the four permeability realizations. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_18.png] view at source ↗
read the original abstract

We present a numerical method for the forward solution of nonlinear partial differential equations (PDEs) in which Bellman-Kalaba quasilinearization reduces the nonlinear problem to a sequence of linear subproblems, each discretized by collocation onto a trial space that is linear in its parameters and solved by a single direct linear least-squares QR factorization. The trial space, which we term Linear-in-Learnables (LiL), comprises representations whose trainable parameters enter linearly, including random-feature extreme learning machines, spectral polynomial bases, and trigonometric expansions, each implemented as a physics-informed neural network. The method thus replaces the nonconvex gradient-based training that limits standard PINNs with a convex per-step solve. We establish local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with the limiting accuracy governed by the best-approximation residual of the trial space rather than by an optimization tolerance. The method, denoted LiL-Q, is assessed on seven benchmarks spanning scalar nonlinear PDEs (Bratu, viscous Burgers, Buckley-Leverett), coupled systems (plane-strain elasticity and the incompressible Navier-Stokes equations in two and three spatial dimensions), and steady-state Darcy flow with heterogeneous permeability. Across these problems, LiL-Q converges in single-digit outer iterations in most cases, even at the coarsest basis sizes and independent of the parameter count. When the exact solution lies in the span of the trial space, the method recovers it to machine precision in a single solve. On the Navier-Stokes benchmarks, it matches or exceeds published PINN solvers with up to two orders of magnitude fewer trainable parameters, without gradient-based optimization.

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 manuscript introduces the LiL-Q method, which applies Bellman-Kalaba quasilinearization to convert nonlinear PDEs into a sequence of linear subproblems. Each subproblem is discretized via collocation onto a Linear-in-Learnables (LiL) trial space (random-feature ELMs, spectral polynomials, trigonometric bases) and solved by a single direct least-squares QR factorization. The paper establishes local Newton-Kantorovich convergence of the outer iteration to a residual-limited neighborhood under an explicit smallness condition, with limiting accuracy governed by the trial-space best-approximation error. Numerical results on seven benchmarks (Bratu, viscous Burgers, Buckley-Leverett, plane-strain elasticity, 2D/3D incompressible Navier-Stokes, heterogeneous Darcy flow) report single-digit outer iterations, machine-precision recovery when the exact solution lies in the trial space, and competitive accuracy versus published PINNs using up to two orders of magnitude fewer trainable parameters without gradient-based optimization.

Significance. If the local convergence result and the smallness condition hold for the reported initial guesses, the work offers a convex, optimization-free alternative to standard PINN training that replaces nonconvex gradient descent with direct linear solves. The explicit separation of approximation error from optimization tolerance and the empirical observation of rapid convergence independent of parameter count are potentially valuable contributions for forward nonlinear PDE problems.

major comments (2)
  1. [Abstract, convergence paragraph] Abstract, convergence paragraph: The local Newton-Kantorovich convergence to a residual-limited neighborhood is asserted under an explicit smallness condition on the initial residual (or guess), yet neither the precise mathematical statement of this condition nor any verification that it is satisfied by the initial guesses and residuals used in the seven benchmarks is supplied. This verification is load-bearing for the claim that the observed single-digit iteration counts follow from the theory rather than problem-specific initialization or basis choice.
  2. [§5, Table 1] §5 (numerical results) and Table 1: The statement that accuracy is governed by the best-approximation residual of the trial space rather than optimization tolerance is central, but the manuscript does not report the computed best-approximation residuals (or their decay with basis size) for the chosen LiL spaces on the benchmarks, preventing direct confirmation that the observed errors are indeed limited by approximation rather than other factors.
minor comments (2)
  1. [§2] The notation for the LiL trial space and the collocation operator could be made more uniform across sections to improve readability.
  2. [§5.3] Figure captions for the Navier-Stokes benchmarks should explicitly state the achieved residual norms alongside parameter counts for direct comparison with the cited PINN references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable comments, which help clarify the presentation of the theoretical guarantees and their relation to the numerical results. We address each major comment below and will incorporate the suggested additions in the revised manuscript.

read point-by-point responses

  1. Referee: [Abstract, convergence paragraph] Abstract, convergence paragraph: The local Newton-Kantorovich convergence to a residual-limited neighborhood is asserted under an explicit smallness condition on the initial residual (or guess), yet neither the precise mathematical statement of this condition nor any verification that it is satisfied by the initial guesses and residuals used in the seven benchmarks is supplied. This verification is load-bearing for the claim that the observed single-digit iteration counts follow from the theory rather than problem-specific initialization or basis choice.

    Authors: The precise mathematical statement of the smallness condition is provided in the Newton-Kantorovich theorem of Section 3. We agree, however, that restating the condition explicitly in the abstract and verifying its satisfaction for the initial guesses employed in the benchmarks would strengthen the link between theory and the reported single-digit iteration counts. In the revision we will (i) insert the exact smallness condition into the abstract convergence paragraph and (ii) add a short verification subsection (or supplementary table) that computes the initial residuals for each of the seven benchmarks and confirms that the condition holds, thereby showing that the observed convergence is consistent with the theorem rather than arising solely from problem-specific choices. revision: yes

  2. Referee: [§5, Table 1] §5 (numerical results) and Table 1: The statement that accuracy is governed by the best-approximation residual of the trial space rather than optimization tolerance is central, but the manuscript does not report the computed best-approximation residuals (or their decay with basis size) for the chosen LiL spaces on the benchmarks, preventing direct confirmation that the observed errors are indeed limited by approximation rather than other factors.

    Authors: We concur that explicit reporting of the best-approximation residuals is necessary to substantiate the claim that the observed errors are governed by the trial-space approximation quality. In the revised manuscript we will compute and tabulate these residuals for the LiL spaces (random-feature ELMs, spectral polynomials, trigonometric bases) used on each benchmark, including their decay with increasing basis size (number of features or polynomial degree). These values will be compared directly with the achieved solution errors in Table 1 and the associated figures, confirming that the errors track the best-approximation bounds rather than any optimization tolerance. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on standard quasilinearization and Newton-Kantorovich theory

full rationale

The paper applies the established Bellman-Kalaba quasilinearization to convert nonlinear PDEs into sequences of linear subproblems, then discretizes each via collocation on a Linear-in-Learnables (LiL) trial space and solves with direct QR least-squares. The local convergence claim invokes the standard Newton-Kantorovich theorem under an explicit smallness condition on the initial residual; this is an external mathematical result, not derived from or equivalent to the paper's own fitted parameters, trial-space choices, or benchmark outputs. No self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations appear in the derivation chain. The seven benchmarks serve as independent numerical validation rather than inputs that force the claimed results by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on quasilinearization reducing the nonlinear PDE to linear subproblems solvable by least-squares and on the LiL property enabling convex direct solves; the smallness condition is a key unverified assumption for the convergence result.

axioms (1)
  • domain assumption Local Newton-Kantorovich convergence holds under an explicit smallness condition on the initial residual or guess
    Stated in the abstract as the basis for the convergence guarantee to a residual-limited neighborhood.
invented entities (1)
  • Linear-in-Learnables (LiL) trial space no independent evidence
    purpose: Representations (random-feature ELMs, spectral polynomials, trigonometric expansions) whose parameters enter linearly to permit convex least-squares solves
    New term introduced in the abstract to group the trial spaces used for the convex per-step solves.

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