Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

Shuiwang Ji; Shurui Gui; Xiner Li

REVIEW 3 major objections 4 minor 20 references

Discovering Physics Laws of Dynamical Systems via Invariant Function Learning

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that shared ODE dynamics can be recovered from environment-biased trajectories by maximizing information under an independence constraint, implemented as the hypernetwork method DIF.

desk verdict New task with a plausible first method, but the central identifiability theorem is not proven as stated—e-independence does not imply independence from the environment function, so the paper's guarantee overreaches. read the letter →

arxiv 2502.04495 v2 pith:DI7UEV57 submitted 2025-02-06 cs.LG

Shurui Gui , Xiner Li , Shuiwang Ji This is my paper · ORCID

classification cs.LG

keywords invariantfunctionlearningdynamicalsystemsordinarydifferentialequationscausaldisentanglementhypernetworkenvironmentindependencescientificequationdiscoverymulti-environmentODEdatasets

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper introduces invariant function learning: given trajectories of the same ODE system observed in environments that differ not just in coefficients but in whole function forms, recover the dynamics shared by all of them — for a pendulum, the frictionless law $-\alpha^2\sin\theta_t$ from damped and powered observations. It argues this recovery is identifiable, not heuristic: in a structural causal model where an invariant function $f_c$ and an environment-specific function $f_e$ compose into the full derivative $f$, the invariant function is the one that maximizes mutual information with $f$ while being independent of the environment. The proposed method, Disentanglement of Invariant Functions (DIF), implements this with a transformer encoder, two disentangling MLPs, a decoder hypernetwork that outputs the predicted derivative function, and an adversarial discriminator that strips environment information from the invariant embedding. On three multi-environment ODE datasets (pendulum, Lotka–Volterra, SIR-epidemic) the recovered invariant functions beat meta-learning and invariant-learning baselines, and symbolic regression converts them into equations close to the true laws. The reason to care is concrete: if correct, it is a route from raw, environment-entangled observations to the underlying physical law.

What carries the argument

The load-bearing object is the invariant function random variable $f_c$, defined by a structural causal model in which $f_c$ and an environment-specific function $f_e$ compose into the observable full dynamics $f$; because $f$ is a collider, conditioning on the observed trajectory entangles the two components and blocks direct fitting. The load-bearing identity is Theorem 3.1, which says $f_c$ is the unique function maximizing $I(h_c(X_p); f \mid X_0)$ under the constraint $h_c(X_p) \perp e$, with Proposition 3.3 converting that information objective into an MSE loss on trajectories. The implementation machinery is a hypernetwork: a transformer encoder maps a trajectory to an embedding, two MLPs separate the invariant and environment parts, a decoder maps the invariant embedding to the parameters of a neural-network derivative function, and an adversarial discriminator enforces the independence constraint. A reference-based hypernetwork implementation that stores parameters by pointer instead of per-sample assignment gives a 16.8x speedup over a naive implementation.

What would settle it

Generate a multi-environment ODE dataset in which the environment-specific term is functionally correlated with the invariant term — for example, one environment adds a force proportional to $\sin(\theta_t)$ with an environment-varying coefficient. The theory predicts DIF still recovers exactly the shared invariant function; if the extracted function instead varies across environments or includes the spurious term, the independence constraint is not sufficient for identification.

Watch

Extended reading notes

Core claim

The paper's central claim is that the invariant function $f_c$ is identifiable from observed trajectories $X_p$ alone, without seeing the environment-free system. Theorem 3.1 states that the true $f_c$ is obtained by maximizing the conditional mutual information $I(h_c(X_p); f \mid X_0)$ subject to $h_c(X_p) \perp e$, and Proposition 3.3 shows this maximization is equivalent to minimizing the mean-square error between the predicted and true invariant trajectories. DIF realizes the principle by splitting the encoded function into an invariant embedding and an environment embedding, decoding the invariant part into a neural-network derivative function, and training adversarially so that a discriminator cannot predict the environment from the invariant embedding. In the reported experiments the extracted function predicts the environment-free trajectory with lower error than the baselines, and symbolic regression on the recovered function yields expressions such as $0.99\,\omega_t$ and $-0.97\,\alpha^2\sin\theta_t$ for the pendulum's natural motion.

Load-bearing premise

The guarantee rests on the assumption that the practical training objective of DIF really enforces the theoretical condition — that the recovered function carries maximum information about the full dynamics while being independent of the environment — since the paper proves that equivalence only in a simplified single-loss case.

Editorial extensions

If this is right

Scientific discovery becomes a single forward pass: a trained DIF network reads one biased trajectory and outputs the shared derivative function, with no per-system fitting or test-time adaptation.
The principle applies to environment changes of whole function form, not just coefficient shifts, so it targets systems where earlier invariant-learning and meta-learning methods were not designed to work.
The recovered invariant function can be handed to symbolic regression, turning a black-box neural predictor into closed-form equations such as $-\alpha^2\sin\theta_t$.
A correct invariant function should make the environment unpredictable, giving a practical diagnostic: if the discriminator can still identify the environment from $\hat{z}_c$, the candidate is spurious.
Because training fits derivatives rather than rolling out an integrator, the method scales to longer trajectories and larger state spaces at training time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the identification theorem is right, 'maximum information under environment-independence' is a general definition of an invariant mechanism that could be transferred to partial differential equations or stochastic dynamics once multi-environment datasets for those domains exist.
The sharpest stress test would be environments whose extra terms are functionally correlated with the invariant term — for example, an environment force proportional to $\sin\theta_t$ with an environment-varying coefficient — which the current datasets may not include; a failure there would bound the identifiability regime.
The framework suggests a measurable signature of non-identifiability: if the invariant embedding's environment-conditional variance stays high, practitioners should distrust the recovered law before symbolic explanation.
The recovered invariant function could serve as an inductive bias for symbolic regression, potentially improving closed-form discovery in systems where direct sparse-regression methods fail because environments entangle the terms.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague—

Know this: the paper's central guarantee, Theorem 3.1, is not proven as stated. The proof in Appx C.2 conflates independence from the environment label e with independence from the environment function fe. A concrete SCM shows the claimed uniqueness fails: take fc ≡ 0, fe = e·W with e ∈ {±1} equiprobable, W ∼ N(0,1), X0 = 0. Then (Xp/t)² = W² is independent of e but carries information about f, and is not fc. So the maximizer of Eq. 2 is not uniquely the true invariant function. This is load-bearing: the abstract's "guaranteed" discovery claim does not follow.

What the paper does well: it identifies a genuinely new task—invariant function learning with environments differing in function form, not just coefficients—that CoDA, LEAD, and MetaphysiCa don't cover. The DIF architecture (transformer encoder, hypernetwork decoder, adversarial independence) is a reasonable first method. The three synthetic ODE benchmarks and released code are useful, and the symbolic regression explanations give a nice sanity check. The empirical advantage over adapted baselines is credible.

Other soft spots are secondary. Prop 3.3 shows the MI-MSE equivalence for a single predictive model, not the actual weighted multi-loss objective, and the independence constraint is applied to the embedding zc rather than the function fc. Evaluation is on simulated data with ground truth generated from the same causal decomposition the method assumes, and there are no comparisons to standard equation discovery baselines. These are addressable; the theorem gap is not a mere technicality.

Who should read this: researchers in scientific ML and invariant learning who want to think about what "invariant function" should mean for dynamical systems. It deserves a serious referee: the task is new, the method is a sensible first attempt, and the empirical story is coherent. But the theoretical claims need to be substantially softened or the proof repaired before the paper can be trusted as a guarantee. I'd recommend sending it to peer review with the expectation of major revision.

Best,

Referee Report

3 major / 4 minor

Summary. The paper introduces invariant function learning (IFL) for ordinary differential equations observed in multiple environments, with the goal of recovering the shared invariant dynamics while discarding environment-specific function forms. The authors propose a causal SCM in which the full derivative function is f = g_comp(fc, fe), design a hypernetwork-based method called DIF that outputs invariant and environment function embeddings, and state Theorem 3.1 claiming that maximizing I(h(Xp); f | X0) subject to h(Xp) ⊥ e uniquely recovers the true invariant function fc. Proposition 3.3 is used to convert this mutual information objective into an MSE loss, and the final objective in Eq. (3) combines derivative fitting for the full and invariant functions with adversarial environment independence. Experiments on three synthetic multi-environment ODE datasets compare DIF against MAML, CoDA, IRM, and VREx, and report both trajectory NRMSE and symbolic-regression explanations.

Significance. The task formulation and benchmark construction are useful: the paper identifies a genuinely different scenario from coefficient-level environment shifts, releases code, and provides a quantitative evaluation protocol via invariant trajectory ground truths. The DIF architecture and the reference-based hypernetwork implementation are practical contributions, and the empirical comparisons show that the proposed method obtains better invariant trajectory predictions than the adapted baselines. However, the central claimed guarantee is load-bearing and, as stated, fails under the paper's own SCM: the uniqueness proof in Appendix C.2 is invalid, and the implemented loss in Eq. (3) is not shown to be equivalent to the theorem's constrained mutual information objective. The paper's value currently rests on the empirical findings plus a plausible but unproven causal framing, not on the stated theoretical guarantee.

major comments (3)

[Appendix C.2 / Theorem 3.1] The uniqueness proof is invalid. Equation (8) treats H(fe) > H(fe|f'_c) as contradicting f'_c ⊥ e, but this only follows if H(fe|f'_c) = H(fe), i.e. if f'_c ⊥ fe. The constraint f'_c ⊥ e does not imply f'_c ⊥ fe because fe = ge(e, εe) can contain an environment-independent noise component εe. A concrete SCM satisfying the paper's Fig. 2 is: fc ≡ 0, e ∈ {±1} with equal probability, W ∼ N(0,1), fe = e·W, f = fe, and X0 = 0. Then h(Xp) = (Xp/t)² = W² is independent of e, carries information about f, and satisfies I(h; f | X0) > 0 = I(fc; f | X0). Thus the true invariant function fc is not the maximizer of Eq. (2), and Theorem 3.1 is false as stated. This invalidates the abstract's claim that discovery of invariant functions is guaranteed by the information-based principle.
[Section 3.2.1 / Eq. (3) / Proposition 3.3] Proposition 3.3 only establishes an equivalence between maximizing I(hθc(Xp); f | X0) and minimizing trajectory MSE for the single forecasting model p(X | hθc(Xp), X0). The practical objective in Eq. (3) is not that objective: it minimizes derivative residuals ∥dX/dt − fc(Xt)∥², adds the full-function derivative loss, and adds discriminator and adversarial losses with separate weights. No argument shows that optimizing this weighted multi-loss objective solves the constrained optimization in Eq. (2). In particular, derivative MSE is not equivalent to trajectory MSE for nonlinear ODEs, and the additional λc, λdis, λadv terms are not derived from the theorem. Consequently, the empirical recovery of fc is not guaranteed by Theorem 3.1 even if that theorem were correct.
[Appendix C.5 / Eq. (3)] The adversarial independence constraint is applied to the hidden embedding zc, while the theorem requires hθc(Xp) ⊥ e. The manuscript does not state or prove that I(e; zc) = 0 implies I(e; fc) = 0; this requires the decoder to be injective and the discriminator to be optimal in a sense that is not established for the finite neural-network implementation. Thus, even under a repaired Theorem 3.1, the practical optimizer could in principle satisfy the constraint on zc while violating it on the function fc that is actually used for invariant forecasting.

minor comments (4)

[Section 1] The sentence 'an important scenario in scientific discovery has been overlooked, i.e., , which aims' contains a missing phrase and a stray comma; it should be reworded.
[Appendix C] Figures 2 and 6 are identical versions of the same SCM; retaining one would reduce redundancy.
[Section 3.2.1 / Eq. (3)] The hyperparameters λc, λdis, and λadv are introduced with a heuristic relationship λadv = λc·λ'adv, but no sensitivity analysis or principled criterion for setting them is provided beyond the random search range in Appendix E.2.2.
[Section 5.2] The invariant ground truth Xc is generated by removing environment terms from the same SCM used in the paper's causal model; this is a reasonable synthetic benchmark, but the text should state explicitly that this evaluation measures recovery of the assumed causal decomposition rather than model-free discovery from raw observations.

Circularity Check

1 steps flagged · score 5.0 of 10

The claimed guarantee that Eq. 2 uniquely recovers fc is not derived from the stated SCM: the proof silently upgrades independence from the environment label e to independence from the environment function fe, so the uniqueness result is partly a restatement of an extra assumption rather than a consequence of the model.

other [Appendix C.2, proof of Theorem C.1 (Thm. 3.1), step leading to Eq. (8)]
"H(f|fc) = H(fe). Similarly, for f′c, we have: H(f|f′c) ≥ H(fe|f′c). Combining these results with Eq. 5, we obtain: H(fe) > H(fe|f′c), which contradicts the independence condition f′c ⊥ ⊥e, as this would require H(fe) = H(fe|f′c)."

By the paper's own SCM (Appx C.1), fe := ge(e, εe) with εe independent of e, so h(Xp) ⊥⊥ e does not imply h(Xp) ⊥⊥ fe. The proof's contradiction needs H(fe|h)=H(fe), i.e. the stronger condition h ⊥⊥ fe, which is never stated or proved. A concrete SCM satisfying Fig. 2 is fc≡0, fe=e·W with e∈{±1}, W∼N(0,1), X0=0, f=fc+fe; then h(Xp)=(Xp/t)²=W² obeys h⊥⊥e and has I(h;f|X0)>0 while I(fc;f|X0)=0, so fc is not the unique maximizer of Eq. 2. Thus the 'guaranteed discovery' reduces to an unstated assumption about the environment function rather than following from the causal model, and Eq. 3 inherits the gap because its adversarial term only penalizes I(e;zc), not dependence on the independent noise εe.

full rationale

The paper's experimental pipeline is self-contained and externally benchmarked: invariant ground truths are generated from known ODE simulators, baselines are adapted meta-learning/invariant-learning methods, and the DIF predictions are evaluated by NRMSE on those benchmarks. There is no load-bearing self-citation and no fitted test label is renamed as a prediction. However, the central theoretical claim is not a clean derivation from the stated assumptions. Theorem 3.1 is the paper's 'guarantee' for invariant function discovery, and its proof treats independence from the environment label e as equivalent to independence from the environment function fe. The SCM explicitly allows fe to contain e-independent noise (εe), so an e-independent statistic of Xp can be informative about fe and about f without being fc. The proof's contradiction therefore assumes the very separation it is meant to establish. This makes the uniqueness/inference result partly definitional: the target fc is characterized in Sec. 2.2.1 as independent of e and most informative about f, and the optimization in Eq. 2 is essentially that same characterization, with uniqueness resting on a hidden stronger condition. Consequently the theoretical guarantee should be read as conditional on an extra, unproved identifiability assumption, not as a consequence of the stated causal graph. The empirical results may still be meaningful, but they are not backed by the advertised proof.

Assumptions & free parameters 3 free parameters · 4 assumptions · 2 invented entities

The paper's central approach rests on a posited causal decomposition of the derivative function into invariant and environment-specific parts, plus the assumption that adversarial training on the embedding achieves the required independence. No external physical constant or independent empirical benchmark is used to validate the decomposition; the only ground truth is synthetically generated from the same decomposition.

free parameters (3)

λc (invariant MSE weight) = sampled from U(1e-7, 1e-4)
Controls how strongly fc alone must predict the full derivative dX/dt; part of the objective Eq 3. The paper searches over this range and reports distributions over hyperparameters.
λadv (adversarial weight) = λadv = λc·λ'adv with λ'adv sampled from U(1e2, 1e6)
Sets the strength of the independence constraint; critical for the method's behavior, tuned by random search.
λdis (discriminator weight) = sampled from U(1e-1, 1)
Weight for discriminator training; less critical but still a hyperparameter in the objective.

assumptions (4)

domain assumption The causal graph in Fig. 2 assumes the derivative function f decomposes as gcomp(fc, fe) with fc and fe independent, and trajectories X are generated by an integrator from f and X0.
This is the central modeling assumption. If real systems do not decompose this way, the invariant function is not identifiable by the proposed method.
domain assumption The true invariant function fc is independent of the environment variable e.
The theory requires fc ⟂ e. The method only approximates this independence adversarially on the embedding zc, not on the function fc itself.
standard math H(f|fc, fe) = 0, i.e., f is a deterministic function of fc and fe with no noise in the composition.
Used in Appx C.2 to derive entropy identities. This is a stated property of the SCM, not a general fact.
ad hoc to paper H(hc(Xp)|X) = 0, i.e., the predicted invariant function is a deterministic function of the observed past trajectory.
Used in the proof of Prop C.3. This is true for the specific hypernetwork implementation but not for arbitrary h, while the theorem is stated for any possible function h.

invented entities (2)

Invariant function fc (latent random variable)
purpose: Represents the shared mechanism across environments, e.g., -α²sinθ for the pendulum.
fc is postulated by the causal graph; no falsifiable prediction outside the paper is provided. The paper only evaluates on synthetic ground truth constructed from the same model.
Environment-specific function fe (latent random variable)
purpose: Captures environment-specific dynamics such as damping, power, or modified interaction terms.
fe is defined as the complementary component of f after subtracting fc. Its existence is an assumption of the SCM, not demonstrated independently.

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Cite this review

Pith. "Pith review of Discovering Physics Laws of Dynamical Systems via Invariant Function Learning." pith.science (2026). https://pith.science/paper/DI7UEV57

@misc{pith2026250204495,
  author       = {Pith},
  title        = {Pith review of: Discovering Physics Laws of Dynamical Systems via Invariant Function Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DI7UEV57}},
  note         = {Machine review of arXiv:2502.04495}
}

abstract

We consider learning underlying laws of dynamical systems governed by ordinary differential equations (ODE). A key challenge is how to discover intrinsic dynamics across multiple environments while circumventing environment-specific mechanisms. Unlike prior work, we tackle more complex environments where changes extend beyond function coefficients to entirely different function forms. For example, we demonstrate the discovery of ideal pendulum's natural motion $\alpha^2 \sin{\theta_t}$ by observing pendulum dynamics in different environments, such as the damped environment $\alpha^2 \sin(\theta_t) - \rho \omega_t$ and powered environment $\alpha^2 \sin(\theta_t) + \rho \frac{\omega_t}{\left|\omega_t\right|}$. Here, we formulate this problem as an \emph{invariant function learning} task and propose a new method, known as \textbf{D}isentanglement of \textbf{I}nvariant \textbf{F}unctions (DIF), that is grounded in causal analysis. We propose a causal graph and design an encoder-decoder hypernetwork that explicitly disentangles invariant functions from environment-specific dynamics. The discovery of invariant functions is guaranteed by our information-based principle that enforces the independence between extracted invariant functions and environments. Quantitative comparisons with meta-learning and invariant learning baselines on three ODE systems demonstrate the effectiveness and efficiency of our method. Furthermore, symbolic regression explanation results highlight the ability of our framework to uncover intrinsic laws. Our code has been released as part of the AIRS library (\href{https://github.com/divelab/AIRS/tree/main/OpenODE/DIF}{https://github.com/divelab/AIRS/}).

Figures

Figures reproduced from arXiv: 2502.04495 by the authors.

**Figure 1.** Multi-environment Pendulum ODE systems. In this example, ODEs with different coefficients and function forms are used to extract their corresponding invariant functions (green). Invariant function learning presents two key challenges. Firstly, invariant mechanisms are not isolated entities, and being intertwined with varying initial conditions, system parameters, and time makes them difficult to define or disentang… view at source ↗

**Figure 2.** Structural causal model. The causal data generation process includes two phases: function generation and trajectory generation, which correspond to our two learning phases in parentheses, namely, function prediction and forecasting. The gray nodes in the causal graph indicate observable variables. 2.2.1. CAUSALITY-BASED DEFINITIONS First challenge: causal formulation. In light of the first challenge, we aim to for… view at source ↗

**Figure 3.** DIF framework. eˆ denotes outputs of the discriminator gϕ introduced in Sec. 3.2.1. Second challenge: function disentanglement. To handle the second challenge of extracting invariant mechanisms in dynamical systems, we aim to define invariant representation in the function space, which requires the access to intermediate functions and the disentanglement formulations. With the two-phase function prediction to fore… view at source ↗

Figures from the paper (12 more)

**Figure 4.** Figure 4: Invariant trajectory prediction errors on 5 methods under 3 multi-environment ODE systems. For each method, we provide model candidates with 80+ random hyper-parameter selections in their searching spaces, i.e., more than 1200 models in the figure. (a) X predictions us…

**Figure 5.** Figure 5: Visualization of trajectory predictions on ME-Pendulum example, on ME-Pendulum, over 75% of our method’s candidates outperform the best results of MAML and CoDA, and more than 93.75% candidates of IRM and VREx. On ME-Lotka-Volterra, the median of our candidates still …

**Figure 6.** Figure 6: Structural causal model. C. Invariant Function Learning foundation C.1. Structural Causal Model In this section, we discuss the trajectory generation process under the Structural Causal Model (SCM) assumption in [PITH_FULL_IMAGE:figures/full_fig_p018_6.png]

**Figure 7.** Figure 7: ME-Pendulum trajectories [PITH_FULL_IMAGE:figures/full_fig_p024_7.png]

**Figure 8.** Figure 8: ME-Lotka-Volterra trajectories. transformation relationship. As shown in Tab. 6, we introduce 4 environments, origin, enlarge, loop, negative. These four environments describe four different models, where some of them are only for math modeling. The origin environment …

**Figure 9.** Figure 9: ME-SIREpidemic trajectories. E. Experimental Details We conduct experiments on 800-sample training sets with a training batch size of 32, which leads to 25 iterations per epoch. For each run, we optimize the neural network with 2,000 epochs, which is equivalent to 50,0…

**Figure 10.** Figure 10: Ablation study on 3 DIF variants under 3 multi-environment ODE systems. For each pipeline, we provide model candidates with 50+ random hyper-parameter selections in their searching spaces, i.e., more than 450 models in the figure. • λc ∼ U(1e − 7, 1e − 4) • λdis ∼ U(1…

**Figure 11.** Figure 11: Trajectory input length study on models trained with different training input length factors under 3 multienvironment ODE systems [PITH_FULL_IMAGE:figures/full_fig_p029_11.png]

**Figure 12.** Figure 12: Environment analysis on models trained with different numbers of training environments under 3 multienvironment ODE systems. F.2. Input Length and Environment Analysis In order to ensure fairness, we fix the input length and the number of environments in our experime…

**Figure 13.** Figure 13: Visualization of trajectory predictions on ME-Lotka-Volterra (a) X predictions using ˆf (b) X c predictions using ˆfc [PITH_FULL_IMAGE:figures/full_fig_p031_13.png]

**Figure 14.** Figure 14: Visualization of trajectory predictions on ME-SIREpidemic One of the major challenges that limits the usage of hypernetworks is the implementation complexity. Most current implementations requires either re-implementing basic neural networks (von Oswald et al., 2020) …

**Figure 15.** Figure 15: Reference-based hypernetwork implementation. In contrast, our vectorized Functional-based implementation leverages PyTorch’s functional methods, achieving 9.2x speedup by avoiding the overhead associated with stateful modules. Note that vectorizing hypernetworks usin…

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Reference graph

Works this paper leans on

20 extracted references · 18 canonical work pages

[1]

Although we constructed multi-environment datasets for ODE systems, extending this to PDEs is significantly more complex and requires domain-specific expertise

Lack of multi-environment PDE datasets: Unlike domain adaptation and generalization tasks, multi-environment datasets for PDE systems are not yet available. Although we constructed multi-environment datasets for ODE systems, extending this to PDEs is significantly more complex and requires domain-specific expertise. Unfortunately, designing such datasets ...

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[2]

Scaling up to PDEs also introduces different training dynamics, which may necessitate additional techniques to stabilize and accelerate training

PDE-specific challenges: Due to the continuous nature of PDEs, applying invariant function learning to PDE systems requires addressing multi-scale and multi-resolution problems. Scaling up to PDEs also introduces different training dynamics, which may necessitate additional techniques to stabilize and accelerate training

work page
[3]

However, this approach does not extend naturally to PDE systems, which would require the development of new interpretability methods tailored to invariant function learning in PDEs

Interpretability challenges: As demonstrated in Appendix F.3, we employ symbolic regression to provide conceptually understandable explanations for sanity checks. However, this approach does not extend naturally to PDE systems, which would require the development of new interpretability methods tailored to invariant function learning in PDEs. Q2: Is fc on...

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Adapted baselines: We detail the design and implementation of adapted baselines from invariant learning and meta-learning frameworks

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Empirical analysis: We conduct extensive empirical studies, including quantitative comparisons, visual analyses, interpretability assessments, and multi-perspective ablation studies on ODE systems (see Appx. F). Q7: What cannot be covered by this paper? What are the limitations of this paper? 16 Discovering Physics Laws of Dynamical Systems via Invariant ...

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fc is a random variable sampled from the structural causal model (SCM) (see Fig

Is fc a deterministic function? No. fc is a random variable sampled from the structural causal model (SCM) (see Fig. 6), obeying the Markov property of graphical causal models. Consequently, all information-theoretic analyses in this work are non-trivial

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X ∈ Rd×T represents a single realization sampled from the matrix-shaped random variable X, i.e., one trajectory

Difference between X and X; f and f. X ∈ Rd×T represents a single realization sampled from the matrix-shaped random variable X, i.e., one trajectory. Similarly, f is a realization of the random function f

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Our notation follows ICLR standards

Other notation-related questions. Our notation follows ICLR standards. Please refer to our notation table (Table 3). It is crucial to distinguish between random variables and their realizations. Q3: What are the differences between coefficient environments and function environments? A: The primary difference lies in which factors vary across multiple environments

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A coefficient environment includes only a single function

Example: Consider the pendulum motion as an example. A coefficient environment includes only a single function. Any change in the rope length or friction coefficient defines a new environment. In contrast, a function environment encompasses all functions that share the same fu...
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Conversely, a function environment consists of multiple functions

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Model design: We propose the first invariant function learning method
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Theoretical analysis: This paper builds upon the well-known invariant learning principle. We establish the foundation of invariant function learning, propose an ODE-based invariant learning structural causal model (SCM) with minimal assumptions, and provide theoretical guarant...
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To systematically evaluate invariant function learning, we construct multiple multi-environment ODE systems

Function environments and datasets: We introduce the concept of function environments. To systematically evaluate invariant function learning, we construct multiple multi-environment ODE systems
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Extending this approach to PDE systems remains an open challenge

Invariant function learning in PDE systems: This work focuses solely on invariant function learning in ODE systems. Extending this approach to PDE systems remains an open challenge. The reasons and current obstacles in this setting have been elaborated in FAQ Q1
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High dimensional system with graph structures: One interesting direction considering graph-like interactions has been partially explored by Shi et al. (2022); Cranmer et al. (2020). These systems are generally nosier and prone to affected by environment effects. Therefore, ext...

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Broader range of applications: Unlike meta-parameters, the learned invariant functions exhibit broader adaptabil- ity. For instance, a discovered physical law can generalize across various systems. Future research could explore applications of invariant function learning, such...
[20]

well performed across all environments

(19) C.5. Theoretical Justification for Adversarial Training To incorporate the independence constraint, we enforce the condition ˆfc ⊥ ⊥e, where ˆfc = hθc (Xp) is the predicted function random variable, not a realization. Since ˆfc ⊥ ⊥e is equivalent to I(e;ˆfc) = 0, and I(e;...

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PMLR, 2017. Giannakis, D. Data-driven spectral decomposition and fore- casting of ergodic dynamical systems. Applied and Com- putational Harmonic Analysis, 47(2):338–396, 2019. Goodfellow, I., Bengio, Y ., Courville, A., and Bengio, Y . Deep learning, volume 1. MIT Press, 2016...

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Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [1]

Although we constructed multi-environment datasets for ODE systems, extending this to PDEs is significantly more complex and requires domain-specific expertise

Lack of multi-environment PDE datasets: Unlike domain adaptation and generalization tasks, multi-environment datasets for PDE systems are not yet available. Although we constructed multi-environment datasets for ODE systems, extending this to PDEs is significantly more complex and requires domain-specific expertise. Unfortunately, designing such datasets ...

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[2] [2]

Scaling up to PDEs also introduces different training dynamics, which may necessitate additional techniques to stabilize and accelerate training

PDE-specific challenges: Due to the continuous nature of PDEs, applying invariant function learning to PDE systems requires addressing multi-scale and multi-resolution problems. Scaling up to PDEs also introduces different training dynamics, which may necessitate additional techniques to stabilize and accelerate training

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[3] [3]

However, this approach does not extend naturally to PDE systems, which would require the development of new interpretability methods tailored to invariant function learning in PDEs

Interpretability challenges: As demonstrated in Appendix F.3, we employ symbolic regression to provide conceptually understandable explanations for sanity checks. However, this approach does not extend naturally to PDE systems, which would require the development of new interpretability methods tailored to invariant function learning in PDEs. Q2: Is fc on...

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[4] [4]

Adapted baselines: We detail the design and implementation of adapted baselines from invariant learning and meta-learning frameworks

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[5] [5]

Empirical analysis: We conduct extensive empirical studies, including quantitative comparisons, visual analyses, interpretability assessments, and multi-perspective ablation studies on ODE systems (see Appx. F). Q7: What cannot be covered by this paper? What are the limitations of this paper? 16 Discovering Physics Laws of Dynamical Systems via Invariant ...

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[6] [7]

fc is a random variable sampled from the structural causal model (SCM) (see Fig

Is fc a deterministic function? No. fc is a random variable sampled from the structural causal model (SCM) (see Fig. 6), obeying the Markov property of graphical causal models. Consequently, all information-theoretic analyses in this work are non-trivial

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[7] [8]

X ∈ Rd×T represents a single realization sampled from the matrix-shaped random variable X, i.e., one trajectory

Difference between X and X; f and f. X ∈ Rd×T represents a single realization sampled from the matrix-shaped random variable X, i.e., one trajectory. Similarly, f is a realization of the random function f

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[8] [9]

Our notation follows ICLR standards

Other notation-related questions. Our notation follows ICLR standards. Please refer to our notation table (Table 3). It is crucial to distinguish between random variables and their realizations. Q3: What are the differences between coefficient environments and function environments? A: The primary difference lies in which factors vary across multiple environments

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