What it shows
An optimiser that changes a package’s geometry needs to know which way each change moves the result. When the electrical and thermal behaviour depend on each other, the result is the point where both agree, and the gradient has to pass through that point rather than through either solve alone.
The record states the result this way:
The published record says, word for word (an excerpt)
Design sensitivities propagate through a self-consistent electro-thermal fixed point by the implicit function theorem, with adjoint-versus-finite-difference agreement of 1.6e-7 in 3-D.
In plain words: the record computes the gradient of the joint answer with an adjoint method, and checks it against the slow way of estimating a gradient, nudging each input and re-solving. In the three-dimensional model the two agree to the relative error quoted above. That agreement, the accuracy of the gradient, is what this page relies on.
Why it matters
Gradient-based design search is only as good as its gradients. A gradient that ignores the coupling between heat and current, or that is computed wrongly through the joint solution, can steer a search the wrong way without any visible error. A check against finite differences is the standard way to catch that.
What is ours, and what is not
Adjoint gradients and differentiating through a fixed point are known methods (see the prior art below). What is ours is their use through this electrical, thermal and mechanical model of a package, and the recorded check.
Who should care
- Package co-design teams who optimise electrical and thermal behaviour together with gradients.
- Reviewers. The record states which of its own figures stand and which it corrected.
The limits, in the record’s words
The published record says, word for word (an excerpt)
THE GRADIENT ACCURACY IS THE STANDING CLAIM. The headline magnitudes were RE-MEASURED DOWNWARD and the invariant claim is the coupled-versus-sequential RELATIVE win, which holds on all 4 specs.
The published record says, word for word (an excerpt)
The 3-D field resolves the Cu heat-pipe path, k_Cu ≈ 400 against glass 1.4, and shrinks absolute ΔT by 1–2 ORDERS versus the 1-D form: at 28 GHz 1.56 → 0.71 °C, against the 1-D 127.6 → 4.9 °C.
The published record says, word for word (an excerpt)
Two further honest flags: 'coupled dominates 4/4' is a ZEROED-OBJECTIVE CONTROL ARTIFACT — the optimizer told to penalize warpage produces less warpage — and 'monotone yield' is linear-by-construction risk with a dead constant of 1.9e-22. State both as definitional, not competitive.
In plain words: the gradient check is the only claim here. The temperature figures in the three-dimensional model are much smaller than in the earlier one-dimensional form, as quoted above, because the model now resolves the copper path that carries heat out, and copper conducts heat far better than glass. The record’s comparison of coupled against step-by-step design is not evidence that coupled design is better: the coupled run was told to penalise warpage and the step-by-step control was not, so it produces less warpage by construction. All of this is simulation of a model, not a measured package.
Open source for this step
Tools and datasets we publish for the package step of building a multi-chip package. They are the checkers around this work, not a copy of the result itself.
- physics-lint: One command that checks a folder of physics models against a fixed set of named physical rules, with findings straight into CI.
- maxwell-lint: Flags a coupling extractor whose answers no passive set of conductors could produce.
- sparam-lint: Is your signal-response model physically possible? Five physical laws checked from the command line.
- interval-core: The interval arithmetic core behind our proofs over whole families of layouts.
- touchstone-tools: Read, write and convert Touchstone files, the standard text files that record how signals pass through a package's connections, and refuse to write one that cannot be read back.
- physics-lint-mcp: The physics checks, callable by an AI agent.
- physics-lint-action: A GitHub Action that fails the build when a model breaks one of a fixed set of named physical rules.
- Signal-response validity corpus: A labelled corpus of physically invalid signal-response networks, and a scorer that grades any checker against it.
- screening-ceiling: The screening-ceiling family as an open dataset.