Mosaic: A Benchmark Suite for Differentiable Physics Solvers

Andrin Rehmann; Dion H\"afner; Heiko Zimmermann

arxiv: 2606.27895 · v1 · pith:CIUT7Q7Snew · submitted 2026-06-26 · ⚛️ physics.comp-ph · cs.LG

Mosaic: A Benchmark Suite for Differentiable Physics Solvers

Andrin Rehmann , Heiko Zimmermann , Dion H\"afner This is my paper

Pith reviewed 2026-06-29 02:10 UTC · model grok-4.3

classification ⚛️ physics.comp-ph cs.LG

keywords differentiable PDE solversbenchmark suitegradient computationfluid dynamicsstructural mechanicsheat transfernumerical conditioningcontainerized solvers

0 comments

The pith

Evaluation of 14 differentiable PDE solvers shows memory limits, numerical stability, and compatibility are bigger barriers than gradient accuracy.

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

Mosaic packages each solver inside a container that exposes a single gradient interface, letting researchers test different solvers on the same problems without rewriting code. The evaluation runs 14 solvers on fluid dynamics, structural mechanics, and heat transfer tasks. It records order-of-magnitude spreads in runtime and in how well-conditioned the Jacobians are, plus cases where a solver cannot even be applied to realistic problem sizes. Solvers that do return gradients nevertheless reach nearly the same final solutions. The work therefore concludes that the main obstacles are resource demands and setup friction rather than the quality of the gradients themselves.

Core claim

Mosaic demonstrates that while differentiable PDE solvers vary widely in computational cost and Jacobian conditioning, all that successfully produce gradients converge to comparable optima, making memory limits, numerical stability, and setup compatibility the primary practical barriers rather than gradient accuracy.

What carries the argument

Tesseract containerization that wraps each solver to provide a uniform gradient API regardless of its original language or automatic-differentiation approach.

If this is right

Order-of-magnitude cost differences make some solvers impractical for given problem sizes.
Structural incompatibilities can rule out entire solvers for realistic tasks.
Gradient-producing solvers reach similar optima across the tested domains.
The dominant obstacles are memory footprint, numerical stability, and integration compatibility.

Where Pith is reading between the lines

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

Developers may begin to publish memory and compatibility profiles alongside accuracy claims for new solvers.
The uniform API could be adopted as a de-facto standard for releasing differentiable physics code.
Training loops that swap solvers mid-experiment become easier to implement and compare.
The same container pattern might be applied to non-PDE simulators that currently lack gradient support.

Load-bearing premise

Wrapping solvers in containers and routing their gradients through a common interface leaves their numerical results and gradient correctness unchanged.

What would settle it

An experiment showing that two solvers reach clearly different optima when both supply gradients would contradict the claim that gradient accuracy is not the limiting factor.

read the original abstract

Differentiable partial differential equation (PDE) solvers underpin solver-in-the-loop ML training, gradient-based optimal control, and inverse problems, yet the practical cost of obtaining correct, usable gradients from a given solver on a given problem is largely undocumented. Integration effort, computational cost, gradient accuracy, and numerical conditioning vary widely across solvers and are discoverable only by trial and error. We introduce Mosaic, an extensible benchmarking framework for differentiable PDE solvers that standardizes access to solver gradients. Each solver is packaged as a containerized component (Tesseract) exposing a uniform gradient API regardless of language or automatic differentiation (AD) strategy, enabling researchers to evaluate, compare, and build on non-trivial physical solvers. Our evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer demonstrates that the benchmark surfaces practically relevant differences: order-of-magnitude variation in computational cost and Jacobian conditioning, alongside structural incompatibilities that eliminate solvers from realistic tasks entirely. Despite this variation, all solvers that produce gradients converge to similar optima, indicating that the practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy alone. Mosaic is open-source and available at https://github.com/pasteurlabs/mosaic.

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

Mosaic packages 14 differentiable PDE solvers into a uniform containerized API and shows big practical differences in cost and compatibility, but the claim that gradients are not the limiting factor rests on unverified packaging fidelity.

read the letter

The paper's core offering is Mosaic, an open-source framework that wraps solvers from different languages and AD backends into Tesseract containers exposing one gradient API. This lets users swap solvers on fluid, structural, and heat-transfer problems without rewriting integration code each time. The evaluation of 14 solvers surfaces real differences: order-of-magnitude spreads in runtime and Jacobian conditioning, plus outright incompatibilities that rule some solvers out for larger tasks.

What stands out is the packaging approach itself. Standardizing access across solvers is a practical step that addresses the trial-and-error integration the abstract describes. The observation that all gradient-producing solvers reach similar optima is worth noting if the numbers hold.

The soft spot is exactly the one the stress-test flags. The conclusion that memory, stability, and compatibility are the real barriers (rather than gradient accuracy) depends on the Tesseract wrappers preserving each solver's original numerics and AD behavior. The abstract gives no detail on how they checked that containerization, language bindings, or unified AD calls did not alter floating-point results, conditioning, or final optima. Without that check, the similarity in optima could be an artifact of the benchmark setup.

This is the kind of work that helps people who are already trying to plug PDE solvers into optimization or ML loops. It is not foundational, but it lowers friction in a niche that is growing. The paper deserves a serious referee because the framework is open and the comparative data, once the verification gap is closed, would be directly usable. I would send it to review with a request for explicit tests showing the packaged solvers match their native versions on at least conditioning and convergence.

Referee Report

1 major / 0 minor

Summary. The paper introduces Mosaic, an extensible benchmarking framework for differentiable PDE solvers. Solvers are packaged as Tesseract containers exposing a uniform gradient API independent of language or AD strategy. An evaluation of 14 solvers across fluid dynamics, structural mechanics, and heat transfer reports order-of-magnitude differences in computational cost and Jacobian conditioning, plus structural incompatibilities that rule out some solvers for realistic tasks; however, all gradient-producing solvers converge to similar optima, leading to the conclusion that practical barriers are memory limits, numerical stability, and setup compatibility rather than gradient accuracy.

Significance. If the central empirical findings hold, the work supplies a practical, reproducible resource for researchers selecting differentiable solvers for solver-in-the-loop training, optimal control, and inverse problems. The open-source release and containerized uniform API are concrete strengths that lower the barrier to comparing non-trivial physical solvers.

major comments (1)

[Abstract] Abstract: the claim that 'all solvers that produce gradients converge to similar optima, indicating that the practical barriers are ... rather than gradient accuracy alone' is load-bearing for the paper's main conclusion, yet the manuscript provides no explicit validation (e.g., side-by-side comparison of native vs. Tesseract-wrapped Jacobians, convergence rates, or final optima on the same test problems) that the containerization and uniform API preserve each solver's original numerical behavior and gradient correctness.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and for highlighting a point that strengthens the manuscript. We address the single major comment below and will revise accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'all solvers that produce gradients converge to similar optima, indicating that the practical barriers are ... rather than gradient accuracy alone' is load-bearing for the paper's main conclusion, yet the manuscript provides no explicit validation (e.g., side-by-side comparison of native vs. Tesseract-wrapped Jacobians, convergence rates, or final optima on the same test problems) that the containerization and uniform API preserve each solver's original numerical behavior and gradient correctness.

Authors: We agree that the claim is central and that explicit validation of numerical fidelity under the Tesseract wrapper would make the argument more robust. The current manuscript reports convergence behavior only for the wrapped solvers; it does not include direct native-versus-wrapped comparisons of Jacobians or final optima. In the revised manuscript we will add a dedicated validation subsection that performs exactly these side-by-side checks on a representative subset of solvers and test problems, reporting both Jacobian differences (where accessible) and convergence trajectories to the same optima. This addition will be placed in the evaluation section and referenced from the abstract. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical benchmark with no derivation chain

full rationale

The paper presents an empirical benchmark suite for differentiable PDE solvers, standardizing access via containerized components and evaluating 14 solvers on metrics like cost, conditioning, and compatibility. No equations, fitted parameters, predictions, or uniqueness theorems are claimed; the central observations (order-of-magnitude variation and convergence to similar optima) are direct empirical results from the benchmark runs, not reductions to self-defined inputs or self-citations. The work is self-contained as a comparison framework without any load-bearing derivation that could be circular.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

This is an empirical benchmark paper with no mathematical derivations, fitted parameters, or new physical postulates; the central contribution is the framework and reported comparisons.

pith-pipeline@v0.9.1-grok · 5748 in / 1125 out tokens · 27131 ms · 2026-06-29T02:10:58.868921+00:00 · methodology

discussion (0)

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L-BFGS+proj and Adam+proj variants apply a solenoidal projection to the gradient Mosaic: A Benchmark Suite for Differentiable Physics Solvers — Rehmann et al., 2026 23 10−4 10−3 10−2 10−1 Relative FD error 2D NS 10−4 10−2 100 3D NS 10−4 10−3 10−2 10−1 Structural 10−4 10−3 10−2 10−1 Thermal 10−2 ε 10−8 10−6 10−4 1 - cos. sim. 10−2 ε 10−7 10−4 10−1 10−2 ε 1...

2026
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and PICT produced no converged result. The failure is structural: the inflow-to-drag map is strongly non-convex (advection-driven, with separation and re-attachment regimes), so secant pairs 𝑠𝑘 =𝑥 𝑘+1 −𝑥 𝑘 taken across iterations sample regions where the local Hessian differs in sign, violating the positive-curvature condition and causing the limited-memo...

2026

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and PICT produced no converged result. The failure is structural: the inflow-to-drag map is strongly non-convex (advection-driven, with separation and re-attachment regimes), so secant pairs 𝑠𝑘 =𝑥 𝑘+1 −𝑥 𝑘 taken across iterations sample regions where the local Hessian differs in sign, violating the positive-curvature condition and causing the limited-memo...

2026