arxiv: 2605.01040 · v1 · submitted 2026-05-01 · 💻 cs.CE · cs.LG

Recognition: unknown

Differentiable Multiphysics Co-Optimization via Implicit Neural Representations: A Transient Hamburger-Cooking Benchmark

Navid Zobeiry

Authors on Pith no claims yet

Pith reviewed 2026-05-09 17:49 UTC · model grok-4.3

classification 💻 cs.CE cs.LG

keywords differentiable optimizationimplicit neural representationsmultiphysics simulationtransient processesco-optimizationsigned distance fieldsEulerian methods

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The pith

Coupling implicit neural geometry to a differentiable Eulerian solver enables end-to-end co-optimization of shape, material state, and process variables across full transient multiphysics rollouts.

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

The paper establishes a framework that represents geometry as a neural signed distance field and links it directly to a compiled multiphysics solver so that gradients can update geometry, boundary conditions, material parameters, and control variables together. It tests the approach on a transient cooking simulation that includes heat transfer, moisture movement, shrinkage, phase changes, and flipping. The method uses continuous relaxations to keep non-smooth events differentiable for backpropagation through time. Results indicate that optimizing geometry alone reshapes the object to ease thermal constraints, while joint optimization spreads improvements across all variables. This matters for design problems where separate treatment of geometry and physics misses their interactions over time.

Core claim

An end-to-end differentiable co-optimization framework is created by representing geometry via Fourier-feature-encoded implicit neural signed distance fields and coupling it to a JAX-compiled Eulerian multiphysics solver. Boundary conditions, initial conditions, process controls, and material parameters are optimized inside the same loop, with continuous relaxations preserving differentiability for reverse-mode automatic differentiation and backpropagation through time. The framework is demonstrated on a transient hamburger-cooking benchmark that incorporates conductive and convective heat transfer, latent energy, moisture and fat transport, shrinkage-induced geometry evolution, evolving and

What carries the argument

Implicit neural representation of geometry as a signed distance field using Fourier feature encoding, coupled to a JAX-compiled Eulerian multiphysics solver with continuous relaxations for non-smooth transitions.

If this is right

Geometry-only optimization produces shapes that reduce thermal bottlenecks even when other parameters stay fixed.
Joint optimization distributes design changes across geometry, material properties, process variables, and boundary conditions through gradients from the full transient simulation.
The same differentiable loop can handle moving boundaries and evolving contact conditions without requiring separate geometry updates.
Competing objectives in multiphysics problems become simultaneously addressable because gradients propagate across the entire time history.

Where Pith is reading between the lines

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

The approach could be applied to other transient systems with moving interfaces, such as additive manufacturing or fluid-structure interaction, where geometry and physics must be tuned together.
It opens the possibility of incorporating real-time sensor data into the optimization loop for adaptive process control.
Extension to three-dimensional problems with more complex contact and phase-change physics would test whether the relaxation scheme scales without introducing new artifacts.

Load-bearing premise

Continuous relaxations can represent non-smooth physical transitions such as phase changes and contact events without losing accuracy or differentiability.

What would settle it

A direct numerical comparison in which the jointly co-optimized design achieves no measurable improvement in the benchmark quality objectives over independently optimized geometry plus parameters, or in which the continuous relaxations produce simulation errors larger than the observed design gains.

Figures

Figures reproduced from arXiv: 2605.01040 by Navid Zobeiry.

**Figure 1.** Figure 1: Overview of the differentiable co-optimization framework. Spatial coordinates are view at source ↗

**Figure 2.** Figure 2: Schematic of the coupled heat-transfer, transport, phase-change, and shrinkage mech view at source ↗

**Figure 3.** Figure 3: Geometry-only optimization trajectory for the frozen ( view at source ↗

**Figure 4.** Figure 4: Final geometry-only solutions for the three initial temperatures: (a) frozen ( view at source ↗

**Figure 5.** Figure 5: Joint optimization trajectory for the refrigerated ( view at source ↗

read the original abstract

The co-optimization of geometry and physical parameters remains challenging in transient multiphysics systems involving moving boundaries, nonlinear material response, phase transitions, and competing objectives. Existing methods often optimize geometry and physical variables separately, rely on simplified steady-state physics, or require offline data generation and reduced design spaces. Here, we present an end-to-end differentiable co-optimization framework that couples an implicit neural representation of geometry with a JAX-compiled Eulerian multiphysics solver. Geometry is represented as a signed distance field using Fourier-feature-encoded spatial coordinates, while boundary conditions, initial conditions, process controls, and material parameters are optimized within the same differentiable loop. Continuous relaxations represent non-smooth physical transitions while preserving compatibility with reverse-mode automatic differentiation and backpropagation through time. We demonstrate the framework using a transient hamburger-cooking benchmark, selected as an interpretable multiphysics problem rather than a culinary optimization exercise. The benchmark combines conductive and convective heat transfer, latent energy effects, moisture and fat transport, shrinkage-induced geometry evolution, evolving contact boundary conditions, flipping-induced boundary-condition changes, and competing quality objectives. Results show that geometry-only optimization modifies shape to relieve thermal bottlenecks, while joint co-optimization distributes the design response across geometry, material state, process variables, and boundary conditions through gradients propagated over the full transient rollout.

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 delivers a working end-to-end differentiable pipeline that jointly optimizes geometry via Fourier INR and multiphysics parameters in a transient JAX Eulerian solver, shown on a hamburger benchmark.

read the letter

The core point is that this framework runs geometry and physics variables through one differentiable loop with backpropagation across the full transient simulation, including moving boundaries and phase changes. They encode geometry as a signed-distance implicit neural field with Fourier features, then feed it into a compiled Eulerian solver that handles conduction, convection, latent heat, moisture transport, shrinkage, and flipping. Continuous relaxations keep the non-smooth transitions compatible with reverse-mode AD. The results illustrate the difference: geometry-only changes just reshape the object to ease thermal issues, while joint optimization spreads adjustments across material state, process settings, and boundaries. That integration is the concrete advance over separate geometry-then-physics workflows or steady-state approximations. The JAX choice keeps the implementation practical for experimentation. The benchmark is chosen deliberately as an interpretable test case rather than a real application, which helps show the effects clearly. The soft spot is the continuous relaxations. The abstract and setup emphasize AD compatibility but give no evidence of conservation checks, comparison to sharp-interface limits, or sensitivity tests on relaxation width. In a problem with multiple phase changes and contact events, any systematic deviation in the smoothed model could let the optimizer find designs that only work inside the relaxed physics. Without those verifications the joint optima remain plausible but unproven. The paper also lacks reported quantitative metrics or baseline comparisons in the summary, so the performance gain over decoupled methods is still qualitative. This is for computational engineers already using differentiable solvers who want to add geometry co-design to transient multiphysics. A reader working on similar optimization loops would pick up the INR-plus-Eulerian pattern and the benchmark construction. It deserves peer review because the pipeline is executable and the benchmark is new, even though the physical fidelity checks need to be added.

Referee Report

2 major / 1 minor

Summary. The manuscript presents an end-to-end differentiable co-optimization framework coupling an implicit neural representation (Fourier-feature-encoded signed distance field) of geometry with a JAX-compiled Eulerian multiphysics solver. Boundary conditions, initial conditions, process controls, and material parameters are optimized jointly in the same loop. Continuous relaxations are used for non-smooth transitions (latent heat, moisture/fat transport, contact boundaries, shrinkage, flipping) to maintain reverse-mode AD and backpropagation-through-time compatibility. The approach is demonstrated on a transient hamburger-cooking benchmark combining conductive/convective heat transfer, phase changes, transport, evolving geometry, and competing quality objectives. Results indicate that geometry-only optimization relieves thermal bottlenecks via shape changes, while joint co-optimization distributes the design response across geometry, material state, process variables, and boundary conditions.

Significance. If the physical fidelity of the relaxations and the quantitative effectiveness of the co-optimization are established, the framework could enable more comprehensive design exploration in transient multiphysics problems with moving boundaries and nonlinear responses. The combination of implicit neural geometry representations with compiled differentiable solvers offers a scalable route to gradient-based co-optimization that avoids separate offline data generation or reduced design spaces.

major comments (2)

[Abstract] Abstract: The central claims about the effectiveness of geometry-only versus joint co-optimization are supported only by qualitative descriptions (e.g., 'modifies shape to relieve thermal bottlenecks' and 'distributes the design response across geometry, material state, process variables, and boundary conditions'). No quantitative metrics, objective-function values, error bars, baseline comparisons, or sensitivity studies are reported, which is load-bearing for assessing whether the gradients over the full transient rollout produce meaningful improvements.
[Method (continuous relaxations)] Continuous-relaxations description (method section): The continuous relaxations for non-smooth physical transitions (latent energy, contact boundaries, shrinkage-induced geometry evolution, flipping) are introduced specifically to preserve compatibility with reverse-mode automatic differentiation. However, no verification is provided against sharp-interface limits, discrete conservation checks, or sensitivity to relaxation width in the Eulerian JAX solver. In a benchmark with multiple coupled non-smooth events, this directly affects whether the propagated gradients yield physically meaningful co-optimized designs or artifacts exploitable by the optimizer.

minor comments (1)

[Abstract] The abstract states the benchmark is 'selected as an interpretable multiphysics problem rather than a culinary optimization exercise'; this framing is clear but could be moved to the introduction for a more formal tone.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make to strengthen the quantitative support and validation of the proposed framework.

read point-by-point responses

Referee: [Abstract] Abstract: The central claims about the effectiveness of geometry-only versus joint co-optimization are supported only by qualitative descriptions (e.g., 'modifies shape to relieve thermal bottlenecks' and 'distributes the design response across geometry, material state, process variables, and boundary conditions'). No quantitative metrics, objective-function values, error bars, baseline comparisons, or sensitivity studies are reported, which is load-bearing for assessing whether the gradients over the full transient rollout produce meaningful improvements.

Authors: We agree that the current abstract and results presentation rely on qualitative descriptions of the optimization outcomes. In the revised manuscript we will augment the abstract and add a dedicated quantitative results subsection that reports objective-function values for the geometry-only and joint co-optimization cases, direct numerical comparisons against the baseline (unoptimized) configuration, error bars obtained from repeated optimization runs with different random seeds, and a brief sensitivity study on key hyperparameters. These additions will allow readers to evaluate the magnitude and statistical reliability of the reported improvements. revision: yes
Referee: [Method (continuous relaxations)] Continuous-relaxations description (method section): The continuous relaxations for non-smooth physical transitions (latent energy, contact boundaries, shrinkage-induced geometry evolution, flipping) are introduced specifically to preserve compatibility with reverse-mode automatic differentiation. However, no verification is provided against sharp-interface limits, discrete conservation checks, or sensitivity to relaxation width in the Eulerian JAX solver. In a benchmark with multiple coupled non-smooth events, this directly affects whether the propagated gradients yield physically meaningful co-optimized designs or artifacts exploitable by the optimizer.

Authors: We acknowledge that explicit verification of the continuous relaxations was omitted from the original submission. In the revised version we will add a new subsection (or appendix) that (i) compares the relaxed model against the corresponding sharp-interface formulation on a simplified 1-D test problem for both energy conservation and interface motion, (ii) reports global discrete conservation errors for mass and energy over the full transient rollout, and (iii) presents a sensitivity study varying the relaxation width parameter while monitoring both the final objective value and the gradient norms. These checks will demonstrate that the chosen relaxation widths produce gradients that remain consistent with the sharp-interface limit within acceptable engineering tolerances. revision: yes

Circularity Check

0 steps flagged

No circularity: optimization outputs are produced by gradient descent, not tautological to inputs

full rationale

The paper describes an end-to-end differentiable co-optimization loop coupling an implicit neural representation (SDF via Fourier features) with a JAX Eulerian multiphysics solver. Geometry, material parameters, boundary conditions, and process variables are treated as free design variables updated via backpropagation through the full transient rollout. Continuous relaxations are introduced explicitly to enable reverse-mode AD, but the reported optima (shape changes relieving thermal bottlenecks, joint distribution of design response) are outputs of the numerical optimization, not quantities defined to equal the inputs by construction. No self-citation chain, uniqueness theorem, or fitted-parameter-renamed-as-prediction appears in the provided text. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review; full details on parameters, assumptions, and any invented constructs are unavailable. The central differentiability claim rests on one explicit domain assumption.

axioms (1)

domain assumption Continuous relaxations can represent non-smooth physical transitions while preserving compatibility with reverse-mode automatic differentiation and backpropagation through time
Explicitly invoked in the abstract as the mechanism enabling end-to-end gradients through the transient rollout.

pith-pipeline@v0.9.0 · 5538 in / 1425 out tokens · 44938 ms · 2026-05-09T17:49:51.565241+00:00 · methodology

discussion (0)

Reference graph

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