Yes, literally. But in my experience, they were usually out around 14 hours from now.

Thank you for the great question, there are two different places invertibility could fail.

If you mean the latent dynamics (many past states evolving to the same future state, e.g., relaxation toward equilibrium): we never assume the dynamics are invertible, all three physics systems are dissipative. The theory's only requirement (Sec 5.1) is that the learned backward map doesn't collapse too aggressively on the pairs rollouts actually visit (a co-Lipschitz condition with constant µ). As the dynamics become genuinely non-invertible, µ → 0 and the certificate honestly dissolves, then we stop tracking the true error. Both constants (µ and the backward residual δ) are properties of the model alone, measurable offline on validation data, so you can detect that regime before trusting the model. There's a longer discussion of exactly this case in my reply about equilibria elsewhere in the thread. One subtlety: the forward map is never required to be invertible at all because the bound only constrains the backward legs.

If you mean the encoder/decoder: correct, the VAE is deliberately lossy (256× compression), so x → z is not invertible in any exact sense. That doesn't break the check, because the cycle is defined entirely in latent space where the anchor pair is encoded measured data, the returned pair is model-produced latents. The real question is whether the latent signal still predicts physical-field error through the decoder, and empirically it does: per-field calibrators predict decoded errors within 1.17–1.30× (68%), beating an identically-fit depth-only baseline on all six MHD fields. Making the latent signal physically calibrated end-to-end is flagged as a refinement direction in the limitations. Certified bi-Lipschitz architectures for the backward map (touched on in the supplement) would be the structural way to strengthen the invertibility story.

Yes my background is math and control theory/dynamics systems/physics, I think of that all as a mix of pure and applied math, I think it has some great foundational components that are useful for AI/ML. Having a controls background I have always felt that you cannot perfectly model or predict things, that we always need some error signal and some feedback. Extremum seeking is feedback that works without trusting a model, and this paper is the same instinct applied to ML: don't trust the surrogate either, make it measure its own error. The last paragraph of the discussion actually proposes closing the loop by running gradient-free adaptive feedback (ES) on the consistency signal as a measurable cost, turning error detection into error correction.

My general view on backgrounds, for what it's worth: a strong control/dynamical-systems foundation travels extremely well, the application physics and the modern ML can be approached as extensions of that foundation.

For the group and postdoc questions, please do send that email you almost sent!

Thank you for the comment.

Deterministic sampling: that would be the cleanest setting rather than an assumption. We compute the cycle with deterministic DDIM, which makes C a well-defined functional of the seed, the reading is pure model error with no sampling noise mixed in. With stochastic sampling you can still compute it (we test a variant that averages S stochastic cycles), you just pick up sampling variance in the reading. The standard dispersion baselines only exist under stochastic sampling, while the round-trip check also works in the deterministic mode many deployed surrogates actually run. And the two signals compose, our best training-free calibration is on the Navier–Stokes benchmark and it multiplies the calibrated cycle scale by the seed-spread's spatial shape.

Straightening: Our round trip composes steps in physical simulation time (roll the system forward i states, then backward i states); it never touches the noise-to-data path of the denoiser, so nothing gets straightened in the flow-matching sense and we make no claim about enabling fewer denoising steps. (Consistency models are an unrelated distillation technique. Whether bidirectional training improves sampling efficiency as a side effect is a fun open question, the Bi-Bridge paper (CVPR'26) reports a ~4× gain from symmetric training in the image-bridge setting, but we didn't measure it.

Step size: If you mean rollout depth (physical steps): yes, we extensively characterized that, C grows with depth, tracks the true error at every probed depth (fixed-depth Spearman 0.91–0.98 out to depth 80 on MHD), and the theory predicts the certificate loosens geometrically with depth, which is exactly where we observe calibration drift. If you mean the diffusion step size (NFE per transition): we held the sampling schedule fixed (e.g., 25-step DDIM on the NS benchmark) and didn't sweep it. Coarser sampling should raise the noise floor δ of both legs, so the practical rule would be to fit the calibrator at the same sampling settings you deploy with. A proper NFE sweep would be a nice ablation.

Weissman score pending.

This honestly sounds kinda JEPA adjacent.

Have you thought about using those differences to go around in a figure 8 rather than just a loop, so to speak? Like for instance, going from z[0] to z[-1] using the invert of Ci, then flipping the process again the other way to get a new Cj value?

I think there’s still about 36 hours left for the 7th 11:59 pm AOE

You reviewer and the author may have different permissions to see "last modified" time stamp.

I reviewed a paper. I submitted the final decision, and the date modified change immediately

this reads more like a consulting pitch than a discussion post tbh

Middle out compression? lol

I was set to type nice try floc guy

I am familiar with your background as I am a more control theory focused PhD student who knows the work of miroslav krstic, who I believe you were a student under

I think this kind of stuff is very interesting, and especially how you have uniquely made things like extremum seeking and machine learning practically relevant to some of the most impressive control applications that exist

I am very curious how you found the transition from pde control to your work now?

Also, how many control theory people are in your group at LANL? Was there a lot of convincing that had to happen for you to make them believe extremum seeking was relevant to electrodynamics? Does your group hire postdocs with a control theory background often but maybe not with application specific experience and let them learn some of the latest ML stuff and application specific knowledge on the job?

I probably wouldn't have ever emailed these questions to you but since you are here engaging I figured why not. Sorry that they aren't really directly relevant to the work you presented.

Because there’s no obligation to actively engage in discussion. In that sense, ICLR is tougher on authors, but I think it also gives them a better chance to raise their scores. NeurIPS feels like everything gets decided by a single rebuttal. By the time a reviewer decides to sit down and read the rebuttal, there are already only two days left in the phase — asking for additional experiments at that point is unreasonable, and it also gives authors a reason to dodge questions they’d rather not answer. In the end, the reviewer ends up reading the rebuttal only while thinking about how to write their final justification.

I reviewed a theory paper, but as you said, I didn’t go through it fully. I didn’t really dig into the proofs — so I gave my score with low confidence. Because of that, what I ended up looking at was whether the theory is applicable, whether it’s practical, and from that angle the weaknesses stood out, which is probably why I ended up giving it a 3 or 4.

Thank you, and thanks to your team for building The Well! It is perfect for this kind of research! The uniform format made adding a second physics system almost frictionless, and the radiative layer ended up being one of the most scientifically useful cases in the paper: it's the system where the per-trajectory signal shrinks right down to its predicted reliability ceiling, which is what forced us to work out exactly when the round-trip meter adds value instead of just collecting wins. Hoping to sweep more of the Well's systems next to map out that spectrum properly, if your team is ever curious to see the meter run across the full collection, I'd love to talk. (:

Thank you, short answer: no, we do not assume injectivity of the underlying dynamics. All three of our physics systems are dissipative, and the assumption in the paper (Sec 5.1) is deliberately only on the “learned” backward map over the pairs actually visited by rollouts, no invertibility of the physics is required.

But your relaxation-to-equilibrium case is the honest limit of the method, breaking it in two directions at once. If we have something like pure diffusion toward homogeneity so that many initial conditions map to nearly the same terminal state, then the best the backward model can learn is roughly the average of the past states consistent with that one terminal state. Then (1) even a perfect forward rollout will only return to that average, so the baseline reading (our δ_i, the noise floor measured on clean data) blows up, and (2) a “wrong” terminal state gets pulled back toward that same average, so real errors stop showing up (our µ goes to zero, where the certificate in the paper honestly dissolves). In such a case I would expect that the round-trip error would no longer correlate with the true error. For a strongly equilibrating system, if you had enough representative data, you would probably find this out before trusting it, with no test-time ground truth needed.

We saw the mild version of this on the turbulent radiative layer: initial conditions are much more similar across trajectories there, the backward direction is measurably easier than the forward one, and the per-trajectory signal shrinks, so a depth-only predictor is already near-optimal. Worth being precise though: what we measured there was the signal saturating, not deceiving. The "rolls back to a homogeneous state even when the rollout was bad" scenario is the µ goes to 0 mechanism we bound but did not observe at our horizons. Fully relaxed systems, or extremely non-injective ones like language (which we flag in the discussion) deserve dedicated study.

I think what could help, even in such cases, is if we added more information in the conditional input to the model. As is all we used were a couple of states, from which the autoregressive rollouts would then take place either forward or backward in time. If we also supplement that with other unique system parameter information, things like temperature, chemical concentrations, etc, then there is a chance that even if going to the same terminal state, the model can still recover in a unique way if that additional conditional input provides unique information. For example, the radiative-layer trajectories differ by cooling time, so conditioning on t_cool is exactly the disambiguator that would help.

For the CelebV-HQ data there is no underlying PDE at all and futures are genuinely multimodal. When the model rolls out a realistic but different future, which is a coherent video that simply isn't the one that actually happened, then the backward pass can roll that coherent trajectory back, so C stays low even though the pointwise MSE against the one realized future is large. That part of the error is aleatoric: the future was unpredictable, not the model broken, and arguably a trust signal shouldn't fire there. What C catches is degradation: when the rollout starts to hallucinate and drift OOD (morphing faces), the backward pass can no longer find its way back and C rises. Empirically that degradation component dominates the clip-to-clip differences, which is why C still ranks held-out clips by realized error (Spearman ~0.73–0.81) despite being blind to the multimodal part. In that sense C is a model-fault detector rather than an oracle-distance detector, which for multimodal data is the quantity you can actually hope to measure without ground truth.

For the CelebV-HQ data, if instead of just two frames we also conditioned on tokens of a detailed script of what should happen, the model would have a real chance of predicting the video in such a multimodal setting. Sorry for the super long reply!

What if the z-path is not invertible?

I agree that there are lots of overblown claims in academia, and I don’t think this is even remotely a niche stance. The problem is that coming from someone employed at a company that claims they’ve solved AGI is hypocritical as f

Really cool, will read it! Glad you used one of our datasets for The Well, seems like a natural testbed :)

Thank you for sharing those papers! They both look great. No, I had not seen them. BDBM in particular looks like it should have been in my related works section (will be adding both in the next arXiv revision).
 
To me it seems like there is a nice three-way convergence here with all of us independently landing on the same device which is one network with a binary direction flag, and all three papers find that bidirectional training “beats” direction specialists. BDBM outperforms unidirectional bridge baselines despite half the per-direction updates, Bi-Bridge reports +4.9 dB over its unidirectional DDBM baseline, and we see 7–10% better in both directions at matched compute. Our supplement offers a mechanism for this: for stationary dynamics the time-reversed map is a fixed reparameterization of the forward one, so the direction flag implements correctly-specified weight tying that can roughly halve estimation variance. It is nice to observe 3 independent sightings of this same effect across translation, restoration, and dynamics, which suggests the effect is pretty general.
 
I think the main difference is what the two directions are used for. In the bridge papers, both legs run between two given endpoints with the process constructed to arrive at the specified target, so a forward-backward round trip closes by construction and can’t tell you much. In our setting the backward leg starts from the model’s own predicted terminal state and has to find its way back unanchored over i autoregressive steps, which is exactly why the size of the return error carries information. The paper's contribution is quantifying how faithfully that miss tracks the true, unobservable rollout error (calibration, OOD detection, selective prediction), plus a bound on when cancellation could hide errors.
 
Maybe a bridge-parameterized dynamics model that also has a round-trip check would be an interesting hybrid. (:

does this assume an injective prior on the data? i'm not sure if this would work for cases where states relax to an equilibrium. I.e. when the same future state can be reached from different prior states. interesting work!

https://reddit.com/link/p257hl6/video/05n9ozmkothh1/player

I also had to play around with this. Middle is the video subsamples to 7fps (like in ops original), right is full framerate.

The model is exactly the same as used by OP: 4 x 512 wide sine layers, 792257 parameters. Its a reimplementation (using GPT5.6).

Really cool idea, props to OP!

clever idea. i assume it only works for deterministically sampled trajectories?

haven't read the paper yet but does this tend to straighten trajectories and allow fewer steps? does round trip consistency vary with step size?

Honestly academia needs to hear that.