arxiv: 2604.17549 · v1 · submitted 2026-04-19 · 🧮 math.NA · cs.NA

Recognition: unknown

Robust Deep FOSLS for Transmission Problems

Alejandro Duque , Paulina Sep\'ulveda , Carlos Uriarte , Jamie M. Taylor , David Pardo

Authors on Pith no claims yet

Pith reviewed 2026-05-10 05:40 UTC · model grok-4.3

classification 🧮 math.NA cs.NA

keywords deep learningleast squarestransmission problemsheterogeneous mediavariance reductionneural networksdiscontinuous solutions

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The pith

A weighted FOSLS formulation lets neural networks solve transmission problems robustly by keeping the loss equivalent to the energy norm regardless of material parameters.

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

This paper develops a deep learning method for transmission problems across material interfaces using a weighted first-order system least-squares approach. The key step is proving that this weighted loss is equivalent to the natural energy norm of the equations, with constants that do not depend on how much the materials differ. This keeps the training objective meaningful even when material contrasts are very large, unlike many standard neural network losses. The same formulation also shows that gradient variance drops automatically as the loss gets smaller. Tests in one and two dimensions with jumps in coefficients confirm that the method handles discontinuities without spurious oscillations when using the ReQU activation.

Core claim

The weighted FOSLS functional is equivalent to a natural energy norm of the transmission problem equations with constants independent of material parameters, and its integral-loss form exhibits passive variance reduction in which gradient variance decreases as the loss value decreases.

What carries the argument

Weighted first-order system least-squares (FOSLS) formulation incorporating an energy-norm Poincaré constant.

If this is right

The optimization remains aligned with a meaningful error measure at arbitrary material contrast.
The gradient estimates exhibit decreasing variance as training loss decreases, unlike VPINNs and Deep Ritz methods.
ReQU activation allows approximation of discontinuous solutions without typical overshoots of smooth networks.
The reduced-order neural network approach with least-squares coefficient computation produces accurate solutions in 1D and 2D interface problems.

Where Pith is reading between the lines

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

If the equivalence holds in practice, the method could be applied to time-dependent or nonlinear transmission problems where parameter independence is critical.
The passive variance reduction might improve training stability in other physics-informed neural network formulations.
Extensions to three dimensions would require verifying that the neural network can still capture the interface behavior effectively.

Load-bearing premise

The neural network architecture and activation can represent the solution well enough for the theoretical equivalence to guide the actual optimization process without introducing new instabilities.

What would settle it

Numerical counterexample where, despite decreasing loss, the computed solution error in the energy norm does not decrease proportionally, or where gradient variance does not reduce with loss.

Figures

Figures reproduced from arXiv: 2604.17549 by Alejandro Duque, Carlos Uriarte, David Pardo, Jamie M. Taylor, Paulina Sep\'ulveda.

**Figure 2.** Figure 2: Initialization of {Φ (j) 1 } 10 j=1 (left) and the corresponding derivatives (right) in 1D. The red markers represent the breaking point of each function. Higher-dimensional initialization. In higher dimensions, the initialization of the coefficients defining Φ1 is extended via tensor products. This construction yields axis-aligned breaking hypersurfaces that are equispaced along each coordinate direction … view at source ↗

**Figure 3.** Figure 3: Initialization of {Φ (j) 1 } 8 j=1 in 2D. In panel (e), the breaking line color corresponds to the associated component. 5. Stochastic Quadrature Since for arbitrary parametric spaces V u θ ×V q θ the integrals present in the loss (18) and in the energynorm Poincar´e approximation (20) cannot be computed exactly, we resort to a quadrature rule QN with N points. The integral of a function I over Ω is then … view at source ↗

**Figure 4.** Figure 4: Loss and H1 0,κ relative errors obtained from the DR functional on a smooth problem for varying number of integration points. Notice the first 100 iterations are not shown. In panel 4a, JRitz denotes the functional (21) with respect to LRitz. 102 103 104 105 106 Iterations 10−3 10−2 10−1 J N = 50 N = 100 N = 200 N = 400 N = 600 (a) FOSLS functional 102 103 104 105 106 Iterations 0.6 0.8 1.0 1.2 1.4 Relativ… view at source ↗

**Figure 5.** Figure 5: Loss and relative errors obtained from the FOSLS functional on a smooth problem for varying number of [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗

**Figure 6.** Figure 6: Comparison between robust and non-robust FOSLS formulation. The relative loss is defined as the square root [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗

**Figure 7.** Figure 7: Relative errors in the energy-norm Poincar´e constant. The reference value is computed by solving numerically [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗

**Figure 8.** Figure 8: Gradient error ∇(u ∗ − uNN) with different neural network architectures. problems that differ in the component of the solution that exhibits less regularity: either the gradient ∇u ∗ or the flux q ∗ . 6.3.1. Transmission problem over a circular interface with discontinuous solution gradient We consider a problem given by a discontinuous coefficient defined across a circular interface: κ = ( 1, |(x, y) − (1… view at source ↗

**Figure 9.** Figure 9: Loss and error evolution and final approximations. In panel (a), we denote [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

**Figure 10.** Figure 10: Relative errors in the energy norms of u and q. For the exact solution, we select u ∗ = sin(2πx) sin(2πy)(|(x, y) − (1/2, 1/2)| 2 − (1/4)2 )/κ. With this setup, the flux q ∗ is everywhere continuous while the gradient ∇u ∗ exhibits a jump discontinuity at the circular interface. We train a neural network for 25 000 iterations [PITH_FULL_IMAGE:figures/full_fig_p019_10.png] view at source ↗

**Figure 11.** Figure 11: Gradient and flux approximations for a problem with a discontinuity in the gradient. Following the energy [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗

**Figure 12.** Figure 12: Directional derivative approximation on different cross-sections and the breaking curves [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗

**Figure 13.** Figure 13: Gradient and flux approximation for a problem with a discontinuity in the flux. Following the energy norms, [PITH_FULL_IMAGE:figures/full_fig_p022_13.png] view at source ↗

**Figure 14.** Figure 14: Approximation of the flux component tangent to the plane interface across different sections and breaking curves [PITH_FULL_IMAGE:figures/full_fig_p023_14.png] view at source ↗

read the original abstract

This work presents a robust, energy-based deep learning framework for solving transmission problems in heterogeneous media, including cases with discontinuous material scenarios. We introduce a weighted First-Order System Least-Squares (FOSLS) formulation involving an energy-norm Poincar\'e constant and prove its equivalence to a natural energy norm of the underlying equations, with constants independent of material parameters. As a result, the optimization landscape remains aligned with a meaningful error approximation even under high material contrast, where standard neural network losses often deteriorate. We further prove that the FOSLS formulation, together with its integral-loss representation, exhibits a passive variance reduction property, whereby the gradient variance progressively decreases as the loss diminishes, in contrast to methods such as VPINNs and Deep Ritz. From a numerical standpoint, we adopt a reduced-order perspective by constructing a low-dimensional space described by a neural network. The optimal coefficients are computed via a least-squares solver, and the space is subsequently improved through gradient-based updates. By selecting the activation function ReQU, the method mitigates the spurious overshoots typically observed in smooth networks when approximating discontinuities. Numerical experiments in 1D and 2D interface settings corroborate these findings.

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.

Desk Editor's Note private letter to a colleague

The paper gives a clean continuous-level theory for a material-independent weighted FOSLS loss plus passive variance reduction, but leaves the discrete neural-network case largely unanalyzed.

read the letter

The key point here is that the authors have come up with a weighted first-order system least-squares loss whose equivalence to the natural energy norm holds with constants that do not depend on the material jumps, and they prove this along with a passive variance reduction property for the integral form of the loss. What they do well is lay out the continuous theory clearly. The equivalence means the loss stays aligned with the actual error even when contrasts are high, which is a common failure mode for other neural network losses. The variance reduction is shown to happen passively as the loss decreases, unlike VPINNs or Deep Ritz. Choosing ReQU activations helps avoid the overshoots that smooth activations produce at discontinuities. Their reduced-order approach, where they solve a least-squares problem for the coefficients and then refine the network via gradients, is a practical way to handle the optimization. The numerical experiments in one and two dimensions with interface problems support the claims for those cases. The main limitation is that all the theoretical guarantees are for the continuous problem. Once the solution is approximated by a neural network, there is no guarantee that the same equivalence constants remain valid or that the variance reduction prevents the training from getting stuck. The paper does not include any analysis of how well the ReQU networks approximate the discontinuous solutions in the energy norm, nor does it test the method on more demanding geometries or higher dimensions. The experiments are corroborative but not exhaustive. This paper will interest researchers working on physics-informed neural networks for heterogeneous media and interface problems. Anyone looking for ways to stabilize training under parameter variation should look at it. It is worth sending to peer review because the ideas are fresh and the mathematical development is substantive enough to benefit from expert feedback.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a weighted First-Order System Least-Squares (FOSLS) formulation for transmission problems in heterogeneous media with discontinuous coefficients. It proves equivalence of this weighted loss to a natural energy norm with constants independent of material parameters, establishes a passive variance-reduction property for the integral-loss representation, and proposes a reduced-order neural-network solver using ReQU activations in which optimal coefficients are obtained by least-squares and the network is refined by gradient updates. Numerical experiments in one and two dimensions are presented to support the claims.

Significance. If the continuous-level equivalence and variance-reduction results hold and are shown to be inherited by the neural-network trial space, the work would supply a theoretically grounded, robust alternative to standard variational losses for high-contrast interface problems. The material-independent constants and the contrast with VPINNs/Deep Ritz on variance behavior are genuine strengths; the hybrid least-squares-plus-gradient reduced-order perspective is also a constructive contribution.

major comments (2)

[Abstract and theoretical sections] Abstract and theoretical sections: the claimed equivalence of the weighted FOSLS to the energy norm with constants independent of the material jump is load-bearing for the robustness statement, yet the manuscript does not demonstrate that these constants remain uniform once the trial space is restricted to the finite-dimensional ReQU neural-network manifold; without this inheritance the optimization-landscape claim for extreme contrasts is not secured.
[Numerical method description] Numerical method description: the passive variance-reduction property is proved at the continuous level, but it is not shown that the property survives the reduced-order optimization (least-squares coefficient solve followed by gradient steps on the network parameters) when the interface geometry is non-trivial or the contrast is large; this gap directly affects the practical reliability asserted in the abstract.

minor comments (2)

[Theoretical formulation] The precise weighting that incorporates the energy-norm Poincaré constant should be written explicitly (including any dependence on the contrast) so that readers can verify independence.
[Numerical experiments] Numerical experiments would be strengthened by reporting the network depth/width, the number of quadrature points, and the precise stopping criterion for the least-squares coefficient solve.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The points raised correctly identify that our theoretical results on equivalence and variance reduction are established at the continuous level, while the neural-network implementation is a reduced-order method. We respond point by point below, clarifying the scope of the analysis and outlining targeted revisions.

read point-by-point responses

Referee: [Abstract and theoretical sections] Abstract and theoretical sections: the claimed equivalence of the weighted FOSLS to the energy norm with constants independent of the material jump is load-bearing for the robustness statement, yet the manuscript does not demonstrate that these constants remain uniform once the trial space is restricted to the finite-dimensional ReQU neural-network manifold; without this inheritance the optimization-landscape claim for extreme contrasts is not secured.

Authors: We agree that the manuscript establishes the equivalence of the weighted FOSLS loss to the energy norm with material-independent constants only at the continuous level. The ReQU network defines a finite-dimensional trial space in the reduced-order solver, where coefficients are obtained by least-squares and parameters are updated by gradients. The paper does not contain a rigorous proof that the same uniform constants hold exactly on this manifold. By the universal approximation properties of ReQU networks in the relevant Sobolev spaces, the constants can be made arbitrarily close to the continuous-case values for sufficiently wide networks, but this is an approximation argument rather than a direct inheritance proof. In the revised manuscript we will add a dedicated remark in the theoretical section discussing the density of the ReQU trial space and its implications for the optimization landscape, together with additional numerical diagnostics (e.g., computed loss-to-energy-norm ratios across increasing network widths) for the highest-contrast test cases. revision: partial
Referee: [Numerical method description] Numerical method description: the passive variance-reduction property is proved at the continuous level, but it is not shown that the property survives the reduced-order optimization (least-squares coefficient solve followed by gradient steps on the network parameters) when the interface geometry is non-trivial or the contrast is large; this gap directly affects the practical reliability asserted in the abstract.

Authors: The passive variance-reduction result is proved for the continuous integral-loss representation. In the reduced-order procedure the least-squares coefficient solve is exact for any fixed network parameters, after which only the network parameters are updated by gradient descent. While we do not supply a theoretical proof that variance reduction is preserved exactly under this hybrid iteration for arbitrary interfaces and contrasts, the numerical experiments already include high-contrast 1-D and 2-D transmission problems with non-trivial interfaces, and the observed optimization trajectories remain stable. In the revision we will augment the numerical section with explicit plots of gradient variance versus iteration number for the 2-D high-contrast example, thereby supplying empirical confirmation that the practical behavior aligns with the continuous-level property. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper's core claims rest on explicit mathematical proofs that the weighted FOSLS formulation is equivalent to a natural energy norm (with material-independent constants) and that the integral-loss form yields passive variance reduction. These steps are derived at the continuous level from the underlying PDE system and are not obtained by fitting parameters to data, redefining quantities in terms of themselves, or importing uniqueness results solely via self-citation. The reduced-order NN discretization is presented as a numerical realization of the already-proven continuous properties rather than a redefinition of them. No load-bearing step reduces by construction to its own inputs, and the derivation remains self-contained against external mathematical benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard functional-analysis results for first-order systems and on the existence of a neural-network approximant with the chosen activation; no new physical entities or ad-hoc fitted constants are introduced.

axioms (2)

domain assumption Existence of a Poincaré constant for the weighted first-order system that is independent of material jumps
Invoked to establish equivalence between the FOSLS loss and the energy norm.
domain assumption The neural network with ReQU activation lies in a space dense enough to approximate discontinuous solutions
Required for the reduced-order training procedure to converge to the true solution.

pith-pipeline@v0.9.0 · 5515 in / 1402 out tokens · 46286 ms · 2026-05-10T05:40:40.094174+00:00 · methodology

discussion (0)

Reference graph

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