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REVIEW 5 major objections 5 minor 30 references

Learning a potential formulation for rate-and-state friction

T0 review · 5 major / 5 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Learned two-potential friction replaces non-variational rate-and-state law and makes implicit time-stepping convex, stabilizing all 77 spring-slider tests the empirical law fails.

desk verdict Promising neural-potential friction model with a real stability win, but the paper's core claim about convex variational implicit updates is untested. read the letter →

arxiv 2507.09796 v1 pith:ABCF52CP submitted 2025-07-13 cond-mat.mes-hall cond-mat.mtrl-sci

Shengduo Liu , Kaushik Bhattacharya , Nadia Lapusta This is my paper · ORCID
classification cond-mat.mes-hallcond-mat.mtrl-sci
keywords rate-and-statefrictionpotentialformulationrecurrentneuraloperatorsimplicittimediscretizationvariationalproblemspring-slidermodelstick-slipinstabilitynetworkpotentials
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

Empirical rate-and-state friction laws, the standard tool for modelling earthquake slip and interface friction, have no potential or variational structure, which makes implicit time-stepping schemes prone to divergence. This paper proposes replacing the empirical law with a potential formulation built from two learned potentials: one depending on slip rate and an internal variable, and one governing the internal variable's evolution. Trained on data generated from the empirical law, the formulation with a single internal variable reproduces rate-and-state friction with a relative error smaller than the empirical law's own fit to laboratory data. In a rate-weakening spring-slider benchmark, the learned potential solved all 77 test cases with both implicit and explicit solvers at every time step tested, whereas the empirical law diverged in most implicit solves and nearly half of explicit solves.

What carries the argument

The load-bearing object is the two-potential ansatz with one hidden internal variable $\xi$. The potentials are $\widetilde{D}^{\dagger}(\dot{x},\xi)$, whose derivative with respect to slip rate gives the friction coefficient, and $D(\dot{\xi})$, the dissipation potential for the internal variable; the Legendre dual $D^*(d)=\sup_{\dot{\xi}\in\mathbb{R}^d}\{\langle d,\dot{\xi}\rangle - D(\dot{\xi})\}$ converts the nonlinear evolution law into the explicit update formula for $\dot{\xi}$. Implicit forward-Euler discretization of the coupled slip-internal-variable system is equivalent to minimizing $$J(x,\xi;\$\Delta$ t) = \tfrac{1}{2}(x-x_p)^2 + \tfrac{1}{2}\bigl((x-2x_n+x_{n-1})/\$\Delta$ t\bigr)^2 + \$\Delta$ t\,\widetilde{D}^{\dagger}\bigl((x-x_n)/\$\Delta$ t,\xi\bigr) + \$\Delta$ $t^{2}$ D\bigl((\xi-\xi_n)/\$\Delta$ t\bigr),$$ and the paper derives Hessian conditions that guarantee convexity when $\Delta t$ is small. The potentials are not hand-specified: they are deep neural networks trained by minimizing the relative $L^p$ mismatch between the model and empirical rate-and-state friction over velocity-jump and continuous-variation histories, an architecture the paper identifies as a recurrent neural operator.

What would settle it

Take the canonical spring-slider with the rate-and-state parameters of Eq. (12), choose a spring stiffness below the critical value at which steady sliding is unstable, and drive it until repeated stick-slip cycles occur; compare the learned potential model against the empirical rate-and-state law using a very fine explicit solve. A divergence in the learned model, or a friction error larger than the $0.0015$ experimental-fit tolerance seen when the empirical law is matched to lab data, would falsify the claim that the two-potential ansatz generalizes to the excluded stick-slip regime.

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Extended reading notes

Core claim

The central claim is that the empirical rate-and-state friction law, which has no potential because of the cross term $V\theta$ in its state evolution, can be emulated by a two-potential state-variable theory. The paper postulates a friction coefficient $f^P = \partial \widetilde{D}^{\dagger}/\partial \dot{x}(\dot{x},\xi)$ and an evolution law obtained through the Legendre transform $D^*$ of a dissipation potential $D$, so that $\dot{\xi} = \frac{dD^*}{dd}\bigl(-\frac{\partial D^{\dagger}}{\partial \xi}\bigr)$. It learns $D^{\dagger}$ and $D^*$ as neural networks from velocity histories generated by the empirical rate-and-state law. With one internal variable, the learned model matches the empirical law with relative $L^2$ error around $5\times10^{-4}$, below the empirical law's fit error of $1.5\times10^{-3}$ to experimental data. The internal variable is shown to be unique up to affine transformations, and the implicit forward-Euler update of a spring-slider system becomes the minimization of a system potential $J$ that is convex for sufficiently small $\Delta t$. Numerically, the potential formulation remains stable in all 77 rate-weakening spring-slider tests, while the empirical formulation diverges in most implicit solves even when the time step is reduced by a factor of 256.

Load-bearing premise

The load-bearing premise is that friction and its history dependence can always be represented by two potentials in one internal variable; the paper assumes this expressivity rather than proving it, and validates it only on slip histories that stop short of fully developed stick-slip, so a failure in that excluded regime would undercut the stability and accuracy claims.

Editorial extensions

If this is right

  • Earthquake-cycle and fault-rupture simulations that stall or diverge with empirical rate-and-state friction can be run with implicit time-stepping as a convex minimization, allowing larger time steps and standard optimization algorithms.
  • The learned friction law can act as a drop-in surrogate: on rate-weakening slip histories consistent with training, its friction error is smaller than the error of the empirical law when fit to laboratory data, so model predictions need not degrade.
  • Since one learned internal variable reproduces a one-state-variable empirical law, the construction offers a data-driven way to discover how many history variables a friction law actually needs.
  • Stability no longer requires shrinking the time step: unlike the empirical formulation, the potential formulation converged in all 77 tested spring-slider cases for every fixed time step from $2^{-13.5}$ s to $2^{-11}$ s.
  • Transfer learning from application-specific velocity histories generated by spring-slider simulations reduces the out-of-distribution friction error from roughly 0.030 to 0.016, indicating the potentials can be adapted to new regimes with modest additional data.

Reading between the lines

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

  • If the two-potential ansatz remains valid in fully developed stick-slip, the regime the spring-slider tests deliberately excluded, the same training pipeline could supply variational formulations for other internal-variable laws such as plasticity, viscoelasticity, and damage, where missing potential structure blocks implicit solvers.
  • The derived Hessian convexity conditions could be turned into a certification tool: evaluate the learned second derivatives and compute a guaranteed safe $\Delta t$ before running a simulation, replacing the paper's empirical stability scan.
  • The result suggests that the lack of a potential in rate-and-state friction is not an intrinsic obstruction but a representational one: a single learned coordinate restores a variational principle, so the apparent non-potential character may be an artifact of the chosen state variable.
  • A direct experimental test is now conceivable: train the potentials on laboratory velocity-stepping data rather than on synthetic empirical-law data and compare stick-slip predictions; the paper notes the current training-data hunger makes this premature, but transfer learning may close the gap.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

5 major / 5 minor

Summary. The manuscript proposes a potential formulation of rate-and-state friction in which the friction coefficient and the evolution of an internal variable are generated by derivatives of two learned potentials D† and D*. The potentials are represented as neural networks and trained on data generated by the empirical rate-and-state law (2)-(3). The training section reports small relative L2 errors with a single internal variable (Table 1) and shows that the learned model generalizes to spring-slider trajectories with larger but reducible error after transfer learning (Table 2). Section 5 derives a variational implicit Euler update and states sufficient convexity conditions, and Section 5.4 reports stability comparisons on 77 spring-slider cases, with the potential formulation stable for all cases and the empirical formulation failing in many implicit solves (Table 3).

Significance. If the numerical claims hold, the proposed potential formulation could improve implicit simulation of rate-and-state frictional interfaces, where the absence of a variational structure is a known obstacle. The paper is clearly written, uses reproducible numerical experiments, and contains a useful calibration of the learning error against the error of fitting the empirical law to experimental data. The demonstration that one internal variable suffices to emulate the empirical law is a valuable result. However, the central computational claim is currently supported by a derivation that is never implemented in the experiments, and the stability tests use off-the-shelf solvers rather than the proposed variational update; the paper therefore overstates what is established.

major comments (5)
  1. [5.4, 5.2] The proposed variational implicit update (17)-(18) is derived but never implemented or run in any experiment. The stability results in Table 3 are obtained with the torchdiffeq package's explicit Runge-Kutta and implicit Adams solvers applied to the neural ODE, not by solving the argmin problem (17). Consequently, Table 3 does not demonstrate that the convex minimization problem is viable, stable, or efficient; it demonstrates that the learned neural dynamics are easier for a generic implicit solver. The manuscript should either implement the variational update and report its performance, or explicitly frame the stability claims as applying to the neural ODE rather than to the derived variational update.
  2. [5.2, Eqs. (21)-(22)] The convexity conditions on J are stated but never verified for the trained potentials. Since D†_NN and D*_NN are known functions, their second derivatives can be computed directly; the statement on page 12 that the stability limits are 'difficult to assess explicitly' because the potentials are 'not known explicitly' is inconsistent with the neural-network representation being available. Without evaluating the Hessian of J (or solving the argmin problem), the claim that the potential formulation 'enables implicit time discretization as a convex variational problem' is unverified. The derivation of conditions (21)-(22) should be given, not merely asserted, and the conditions should be checked on the trained potentials for the time steps used in Section 5.4.
  3. [2, 3] The claim that empirical rate-and-state friction has no potential formulation is only sketched: the paper states that the Vθ cross term in the evolution law prevents the existence of potentials. This argument does not rule out the general two-potential, internal-variable class defined by Eqs. (5)-(6), which is the class being proposed. Since this distinction is a central motivation for the work, a rigorous argument (or a precise theorem identifying the class of potential formulations that the empirical law fails to belong to) is needed, or the claim should be substantially weakened.
  4. [5.3-5.4] The reported accuracy and stability advantages are demonstrated only in regimes that exclude fully developed stick-slip, which is precisely the rate-weakening regime where empirical rate-and-state friction is numerically challenging. Section 5.3 explicitly restricts the loading trajectories to avoid 'fully developed stick-slip in which the sliding velocity changes over several orders of magnitude', and the training velocity range is V/V* ∈ [10^-3, 10^1]. The 77 test cases in Section 5.4 should be characterized in terms of their slip-rate ranges and stick-slip content; if the range is similarly limited, the conclusions should be qualified accordingly. Testing on cases with larger velocity contrast, or at least reporting the velocity range of the 77 cases, is necessary to support the claimed improvement in the rate-weakening regime.
  5. [5.2, Eqs. (18)-(19)] The variational functional is dimensionally inconsistent with the equation of motion. Equation (14) contains the mass m, the spring stiffness k, and the gravitational factor mg, but the potentials in Eqs. (18)-(19) are written as Esp(x) = (1/2)(x - xp)^2 and Ein(x) = (1/2)((x - 2xn + xn-1)/Δt)^2, with no m, k, or mg. Equation (21) later reintroduces m. As written, the stationarity of J does not reproduce Eq. (16) unless m = k = mg = 1, which is not stated. The authors should either include the missing coefficients in Eqs. (18)-(19) or state clearly that a nondimensionalization with these values has been adopted.
minor comments (5)
  1. [Figure 4] The axis labels in Figure 4 contain '×10□7' and '×10□8' with unrendered superscripts; these should be typeset correctly.
  2. [Table 3 and Section 5.4] The caption of Table 3 should identify explicitly which solver is used for the NN and RS rows; the current phrase 'implicit, explicit solvers' is ambiguous.
  3. [Algorithm 1] In Algorithm 1, the line computing f_NN contains a stray bracket ('f_NN_i ]'), and the loss line mentions only w_D† and w_D while the updates should also refer to w_D*; please correct the notation.
  4. [Section 4.2] The sentence 'We train for 100 epoch' should read 'We train for 100 epochs'.
  5. [References] Reference [7] appears to be mislabeled for the Dieterich 1981 work; please verify the bibliographic details.

Circularity Check

2 steps flagged · score 4.0 of 10

The core emulation result is held-out fitting rather than circular, but the uniqueness proof is deferred to an in-preparation self-citation and the headline stability tests use transfer-trained potentials.

  1. uniqueness imported from authors [Section 4.4, paragraph beginning 'Let us consider the uniqueness of the internal variable.']
    "However, an underlying assumption in our potential formulation is that D is only a function of ˙ξ. In such a setting, under some hypothesis, it is possible to show (see [3] for the rigorous argument) that the internal variable is unique up to an affine transformation. ... Thus, the internal variable ξ is unique up to affine transformations."

    The uniqueness statement is presented as a result of this paper ('We can show ...'), but the rigorous argument is deferred to reference [3], an in-preparation companion by the same research group (Bhattacharya, Raj, and Stuart). The in-paper empirical check, three trainings with a linear-regression coefficient above 0.98, is suggestive but does not prove uniqueness. The paper's conclusion that the internal variable is canonical up to affine transformations therefore rests on a load-bearing self-citation whose content is not independently established in the present manuscript.

  2. fitted input called prediction [Section 5.3, transfer-learning paragraph; Section 5.4, stability table using the enhanced potentials.]
    "To confirm that the larger errors are due to the out-of-distribution tests, and not due to deficiencies of the formulation, we create a new data set using 200 velocity profiles generated by the solution of the spring-slider system using the RS model. We use transfer learning following [30]: we initialize the neural networks using the parameters obtained in Section 4.2, and then further optimize it over this new data set. ... We use these enhanced potentials in the following section."

    The numerical-stability comparison in Table 3, which supports the paper's headline claim of superior stability for the potential formulation, is run with the 'enhanced' potentials that were additionally optimized on 200 spring-slider-like velocity trajectories generated from the same empirical RS model. The stability test is therefore not a pure evaluation of the originally trained potential structure; the model has already been fitted to the application domain being tested. While stability itself is not a training label, using the transfer-trained NN' for the stability claim makes part of the reported advantage a post-hoc fitted result rather than an out-of-distribution prediction.

full rationale

The paper's central learning-to-emulate claim is not circular in a strict sense: the test sequences in Table 1 and Figure 2 are held out from training, so the small reported errors are genuine independent evidence that the neural potentials can reproduce the empirical rate-and-state law on unseen velocity histories. Similarly, the derivation of the variational update and convexity conditions in Eqs. (17)-(22) is a mathematical construction from the proposed two-potential ansatz, not a re-statement of the training data. The paper does not implement the derived argmin update in the stability experiments, and it never verifies conditions (21)-(22) on the trained networks; that is a verification gap, not a circular reduction. However, two load-bearing elements do weaken the independence of the stated results. First, the uniqueness of the internal variable is justified by citation to an in-preparation companion paper by the same authors, making the 'unique up to affine transformations' claim depend on unpublished self-citation. Second, the stability comparison that forms the main practical conclusion uses potentials that have been transfer-learned on spring-slider trajectories from the empirical RS model, so the favorable stability numbers are partly a consequence of additional fitting to the target domain. Taken together, these issues warrant a moderate circularity score of 4: the central emulation and variational derivation retain independent content, but parts of the claimed theoretical and numerical payoff reduce to self-citation or domain-specific fitting.

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

The central construction rests on a postulated two-potential ansatz, an unproved impossibility assertion about the empirical law, and a uniqueness result taken from an unavailable companion paper. The free parameters are the neural network weights and the chosen latent dimension; the internal variable is a learned construct without independent empirical handle.

free parameters (3)
  • Neural network weights w_D† and w_D* = Not enumerated; trained via backpropagation to minimize Eq. (11)
    The potentials are parameterized by deep neural networks and fit to match friction outputs of the empirical RS law. All predictive content of the surrogate depends on these fitted weights.
  • Optuna hyperparameters (depth, width, p, batch size) = D* has 5 hidden layers [1024, 1024, 1024, 1024, 64]; D† has 6 hidden layers [512, 512, 128, 128, 16, 1024]
    Selected on training and validation data via Optuna; these choices affect the fit and are not derived from theory.
  • Internal variable dimension = 1
    Chosen by comparing errors for dim(ξ)=0, 1, 2; one internal variable matches the one state variable in the empirical law.
assumptions (3)
  • domain assumption A history-dependent friction coefficient can be represented by two potentials D†(ẋ, ξ) and D(ξ̇) via Eqs. (5)-(6).
    Postulated in Section 3; no existence theorem is given, and the claim is validated only by numerical fitting to data generated from the empirical law.
  • ad hoc to paper The empirical rate-and-state friction has no potential formulation because of the Vθ cross term in the evolution law.
    Stated in Section 2 without proof. It motivates the entire approach, but if false, the impossibility motivation weakens while the numerical surrogate could still work.
  • ad hoc to paper Uniqueness of the learned internal variable up to affine transformation (cited to [3], in preparation).
    Section 4.4 relies on the in-preparation companion paper [3] for the rigorous argument; that manuscript is not available for independent checking.
invented entities (2)
  • Internal variable ξ (learned latent state)
    purpose: Replaces the empirical state variable θ; its evolution is determined by the Legendre-transformed potential D*.
    Discovered from data during training rather than measured independently. The paper shows uniqueness up to affine transforms, but no physical handle outside the fitted model is provided.
  • Neural network potentials D†_NN and D*_NN
    purpose: Surrogates for exact potentials; their derivatives generate the friction coefficient and the state evolution.
    Mathematical constructs with parameters fit to data; they make no falsifiable prediction outside the paper's own training regime.

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Cite this review

Pith. "Pith review of Learning a potential formulation for rate-and-state friction." pith.science (2026). https://pith.science/paper/ABCF52CP

@misc{pith2026250709796,
  author       = {Pith},
  title        = {Pith review of: Learning a potential formulation for rate-and-state friction},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ABCF52CP}},
  note         = {Machine review of arXiv:2507.09796}
}
read the original abstract

Empirical rate-and-state friction laws are widely used in geophysics and engineering to simulate interface slip. They postulate that the friction coefficient depends on the local slip rate and a state variable that reflects the history of slip. Depending on the parameters, rate-and-state friction can be either rate-strengthening, leading to steady slip, or rate-weakening, leading to unsteady stick-slip behavior modeling earthquakes. Rate-and-state friction does not have a potential or variational formulation, making implicit solution approaches difficult and implementation numerically expensive. In this work, we propose a potential formulation for the rate-and-state friction. We formulate the potentials as neural networks and train them so that the resulting behavior emulates the empirical rate-and-state friction. We show that this potential formulation enables implicit time discretization leading to efficient numerical implementation.

Figures

Figures reproduced from arXiv: 2507.09796 by the authors.

Figure 1
Figure 1. Training of D† and D through fitting f NN to f RS . with parameters w = {wD† , wD∗ } and we train the networks using data generated from the empirical rate-and-state formulation. We select a set of velocity histories, {Vi(t), t ∈ [0, T]} N i=1, and calculate the resulting friction coefficients, {f RS i } N i=1, from the empirical rate￾and-state laws (2,3) and {f NN i ]} N i=1 from the proposed model (5,9,10). We the… view at source ↗
Figure 2
Figure 2. Two typical test examples of data {Vi , f RS i } and results of the trained RNO {f NN i }, one from the velocity jump sequence (upper row) and one from a continuous variation sequence (lower row). Loss is relative L2, (11), and there is one internal variable in the RNO. increasing the number of internal variables beyond one does not lead to better results. The fact that it suffices to use dim(ξ) = 1 makes intuitive … view at source ↗
Figure 3
Figure 3. A typical fit of empirical rate-and-state formulation [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Learnt potentials (a) D∗ ( ˙d) and (b) D† . D∗ is convex, but D† is not convex in ( ˙x, ξ). The red dots show the trajectories of velocity-jump dataset, while the green dots show the trajectories of continuous variation dataset. smooth, such that η = g(ξ). We can show …
Figure 5
Figure 5. Figure 5: Comparison of the internal variables obtained from three models trained using [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
Figure 6
Figure 6. Figure 6: Example: spring slider under displacement-control driving force. [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: An example of spring-slider simulations with the empirical rate-and-state friction [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Growth of relative L2 error in ˙x(t) as ∆t increases. ∆t [s] 2−13.5 2 −13.0 2 −12.5 2 −12.0 2 −11.5 2 −11.0 Mean Relative L2 error NN, implicit 5.993e-06 3.636e-06 4.807e-06 4.716e-06 6.282e-06 8.508e-06 NN, explicit 6.130e-06 3.808e-06 4.786e-06 4.397e-06 5.968e-06 7.…

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