{"id":"025b20d3-9372-4803-b62d-a9362c10e015","arxiv_id":"2507.09796","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A machine-learned potential formulation matches empirical rate-and-state friction and makes implicit time stepping a convex optimization problem.","lead":"This paper trains neural network potentials to reproduce rate-and-state friction, turning implicit time steps into a convex minimization problem. The result is a friction surrogate that stayed stable in spring-slider tests where the standard empirical law diverged.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The convex variational implicit update at the center of the claim is derived but never implemented; the stability experiments use an off-the-shelf Adams solver, and convexity of the trained potentials is never verified.","rationale":"The reader's weakest assumption concerns the expressiveness of the two-potential ansatz with one internal variable. That is a real concern, but the paper does provide empirical evidence for expressiveness on held-out velocity histories and on spring-slider trajectories, albeit with larger error and without fully developed stick-slip. My stress-test focuses on a more direct gap: the mechanism that supposedly delivers the headline benefit - the convex variational implicit update - is never exercised in the numerical stability experiments. The stability comparison uses generic ODE solvers, so what is tested is the learned potential-based ODE, not the variational formulation. The convexity conditions are derived but not verified for the trained potentials at the chosen time steps. This matters because the paper's abstract and strongest claim promise efficient implicit solution through a convex variational problem; without implementing that update, the claim is under-supported. I do not think this invalidates the paper; it is an addressable gap that should be closed before the central claim is fully accepted. The reader's conditional verdict already captures the need for clarification, so I recommend no change in the verdict, but the concrete test above should be a condition for full acceptance. I also note the paper's own limitations: the no-potential assertion for rate-and-state friction is brief, the uniqueness proof is deferred to an in-preparation reference [3], no code or data are shipped, and the accuracy comparison to experiments is logically loose. These are secondary to the untested variational update but reinforce the conditional stance.","tokens_in":11919,"tokens_out":5178,"duration_ms":66632,"concrete_test":"Implement the implicit update by numerically minimizing J(x, xi) in Eq. (17)-(18) with the trained single-internal-variable potentials, using L-BFGS or Newton-CG, and run all 77 spring-slider cases at the time steps listed in Table 3. Report convergence rates and solution quality against the Adams results. In parallel, evaluate the Hessian of J (or the sufficient conditions (21)-(22)) at the states visited by these simulations: if the minimization fails for some cases or the Hessian is indefinite at the chosen Delta t, the convex-variational claim is not supported by the reported experiments.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the potential formulation leads to a convex variational problem for implicit time discretization, enabling stable and efficient implicit computations. Section 5.2 derives the variational update (17)-(18) and gives sufficient convexity conditions (21)-(22), but Section 5.4 tests stability using the off-the-shelf solvers available in torchdiffeq: explicit 4th-order Runge-Kutta and implicit Adams. The proposed argmin update is never implemented or run in any experiment. Consequently, the reported superior stability of the NN formulation is evidence that the learned neural ODE is easier for a generic implicit solver, not that the derived convex minimization problem is viable. Moreover, the convexity conditions involve the second derivatives of the trained neural-network potentials, which are not checked at any of the time steps used in Table 3; the paper explicitly states that the stability limits are difficult to assess because the potentials are not known explicitly. The load-bearing logical step 'this potential formulation enables implicit time discretization as a convex variational problem' is therefore supported only by an unverified derivation, not by the numerical evidence presented. Since the stability tests also avoid fully developed stick-slip (Section 5.3), the claim that the formulation emulates rate-and-state friction in the difficult rate-weakening regime remains untested at the precise point where the potential formulation is claimed to provide the largest benefit.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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).","tokens_in":12232,"tokens_out":5618,"duration_ms":64182,"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":[{"comment":"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.","section":"5.4, 5.2"},{"comment":"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.","section":"5.2, Eqs. (21)-(22)"},{"comment":"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.","section":"2, 3"},{"comment":"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.","section":"5.3-5.4"},{"comment":"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.","section":"5.2, Eqs. (18)-(19)"}],"minor_comments":[{"comment":"The axis labels in Figure 4 contain '×10□7' and '×10□8' with unrendered superscripts; these should be typeset correctly.","section":"Figure 4"},{"comment":"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.","section":"Table 3 and Section 5.4"},{"comment":"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.","section":"Algorithm 1"},{"comment":"The sentence 'We train for 100 epoch' should read 'We train for 100 epochs'.","section":"Section 4.2"},{"comment":"Reference [7] appears to be mislabeled for the Dieterich 1981 work; please verify the bibliographic details.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a promising early version, but the central computational claim is not tested: the proposed variational update is not implemented, and the convexity conditions are not verified. The paper also defers a key uniqueness proof to a companion paper currently in preparation [3]. A major revision that implements the variational update, checks the Hessian conditions, and either extends the stability tests toward the stick-slip regime or qualifies the claims would substantially strengthen the paper. The current version is not ready for acceptance as is."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper. First, it has a genuinely interesting idea: model rate-and-state friction with two learned dissipation potentials and an internal variable, then use the potential structure to get a variational implicit update. Second, the paper never actually runs that update. The stability experiments in Table 3 use off-the-shelf Adams and RK4 from torchdiffeq, not the argmin formulation in equations (17)-(18). So the headline claim \"this potential formulation enables implicit time discretization as a convex variational problem\" is supported only by a derivation, not by any numerical evidence.\n\nWhat is new and good: the two-potential ansatz (5)-(6) is a sensible specialization of the potential-RNO framework to friction. The training is careful, with a reasonable comparison: the learned model fits the empirical rate-and-state law with relative L2 error around 5e-4, an order of magnitude below the empirical law's own fit to the experimental sequence in Figure 3. The finding that one internal variable is enough is consistent with the single state variable in the target law, and the uniqueness evidence (three models, linear regression >0.98) is decent. The stability result—NN stable in all 77 spring-slider cases at all time steps, while RS diverges in roughly half to two-thirds of cases—is striking and would be useful even if the mechanism turns out to be different from the variational one.\n\nThe soft spots are in proportion to how much the claims rely on them. The biggest one is the unvalidated variational update. You could implement the argmin minimization easily and check convexity conditions (21)-(22) on the trained potentials at the actual time steps; the paper says this is difficult \"since we do not know the potentials explicitly,\" but that is exactly what automatic differentiation is for. Second, the no-potential proof for empirical RS is only a sketch (\"easily verified ... due to the cross term Vθ\"), which is insufficient if that claim is load-bearing. Third, fully developed stick-slip, where the rate-weakening instability is strongest, is excluded from the accuracy test, and the stability test does not clearly report whether any of the 77 cases actually entered that regime. Minor issues: no code or data shipped, and the Hessian conditions are stated without derivation.\n\nOverall, this is a credible paper with a clear direction. The gap is addressable and the authors seem aware of some limitations, but the central claim needs an actual implementation of the variational update before it can be taken as demonstrated.\n\nI would send it to peer review—the idea deserves referee time—but I would ask for the missing experiment and a released implementation before acceptance. For a reading group, it's worth a session.","headline":"Promising neural-potential friction model with a real stability win, but the paper's core claim about convex variational implicit updates is untested.","tokens_in":12720,"tokens_out":4689,"would_cite":true,"duration_ms":52112,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["rate-and-state friction","potential formulation","recurrent neural operators","implicit time discretization","variational problem","spring-slider model","stick-slip instability","neural network potentials"],"falsifier":"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.","tokens_in":11705,"feed_emoji":"⚙️","tokens_out":9776,"duration_ms":105136,"temperature":0.7,"pith_summary":"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.","feed_headline":"Learned friction potentials keep all 77 fault-slider runs stable","feed_subtitle":"By encoding rate-and-state friction as two neural-net potentials, implicit time-stepping becomes a convex problem.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the empirical rate-and-state friction law and the state evolution equation that the learned potentials are trained to emulate.","marker":"[6]"},{"why":"Defines the state-variable friction laws and evolution laws, and motivates the velocity-jump test sequences used to generate training data.","marker":"[27]"},{"why":"Establishes the stability analysis and ill-posedness of rate-weakening steady sliding that motivates seeking a variational potential formulation.","marker":"[26]"},{"why":"Introduces recurrent neural operators for history-dependent behaviour and the velocity-history data-generation strategy adopted in this work.","marker":"[15]"},{"why":"Provides the continuous-time recurrent neural operator architecture with learned internal variables that the potential ansatz extends.","marker":"[2]"},{"why":"Companion paper cited for the rigorous proof that the learned internal variable is unique up to an affine transformation.","marker":"[3]"},{"why":"Supplies experimental friction data whose empirical-law fit error of 0.0015 sets the reference for judging the learned model's accuracy.","marker":"[14]"},{"why":"Defines the spring-slider stability regimes, including the critical stiffness and stick-slip behaviour, used to choose the 77 benchmark loading cases.","marker":"[10]"}],"fun_headline_variants":["Neural friction potentials tame all 77 fault-slider tests","Friction without potential? Learned model fixes implicit solves","Two learned potentials make rate-state friction implicit-solver friendly","All 77 fault-sliders stable with learned friction potentials","Rate-state friction gets a potential via neural nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Neural friction potentials tame all 77 fault-slider tests","Friction without potential? Learned model fixes implicit solves","Two learned potentials make rate-state friction implicit-solver friendly","All 77 fault-sliders stable with learned friction potentials","Rate-state friction gets a potential via neural nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000572,"raw_usage":{"total_tokens":2712,"prompt_tokens":963,"completion_tokens":1749,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":579,"completion_tokens_details":{"reasoning_tokens":1684}},"tokens_in":579,"tokens_out":1749,"duration_ms":13002,"temperature":1.0,"reasoning_tokens":1684,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:47:15.750309+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Dieterich","cited_arxiv_id":null,"evidence_quote":"Supplies the empirical rate-and-state friction law and the state evolution equation that the learned potentials are trained to emulate."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the state-variable friction laws and evolution laws, and motivates the velocity-jump test sequences used to generate training data."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the stability analysis and ill-posedness of rate-weakening steady sliding that motivates seeking a variational potential formulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces recurrent neural operators for history-dependent behaviour and the velocity-history data-generation strategy adopted in this work."},{"cited_title":"Bhattacharya, B","cited_arxiv_id":null,"evidence_quote":"Provides the continuous-time recurrent neural operator architecture with learned internal variables that the potential ansatz extends."},{"cited_title":"Bhattacharya, M","cited_arxiv_id":null,"evidence_quote":"Companion paper cited for the rigorous proof that the learned internal variable is unique up to an affine transformation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies experimental friction data whose empirical-law fit error of 0.0015 sets the reference for judging the learned model's accuracy."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the spring-slider stability regimes, including the critical stiffness and stick-slip behaviour, used to choose the 77 benchmark loading cases."}],"review_version":1}