{"id":"3c1300de-4189-4503-b1ff-6d234a1fa7d1","arxiv_id":"2506.13061","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A TV convergence bound O(d^{7/4} ε^{1/2} + d(dH)^p) is proved for p-th order (exponential) Runge-Kutta samplers of probability-flow ODEs under C² smoothness of the learned score.","lead":"This paper proves total-variation convergence bounds for high-order Runge-Kutta solvers of diffusion probability-flow ODEs, including exponential variants used by fast samplers such as DPM-Solver. If correct, it delivers the first p-th order sampling guarantee that needs only bounded first and second derivatives of the learned score, under smooth noise schedules.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption 3.5's uniform sup_x bounds on learned-score derivatives are load-bearing, and Section 6.1 verifies only sample maxima, not a true supremum over R^d.","rationale":"The paper's proof is systematic, and the Gaussian-mixture experiments genuinely support the H^p discretization rates and the non-compounding of score and temporal errors; I credit that as real evidence for the structural part of the analysis. My concern is narrower but load-bearing: the score-error term and map-invertibility step in Theorem 3.7 depend on uniform derivative bounds for the learned network, and the only evidence for those bounds is the heuristic visualization in Section 6.1, which does not compute a supremum over R^d. The reader's weakest_assumption correctly identifies Assumption 3.5; I agree. I also considered whether the compressed proof of Theorem 4.2 for exponential Runge–Kutta schemes is a more serious gap, but the Appendix A expansion plus Lemma 5.7 likely can be filled, whereas the sup_x verification has no described method at all. Notably, the convergence tests in Section 6.2 do not validate Assumption 3.5 either, because the artificial score error δ(x1) has discontinuous second derivative, so the test model is not C². For these reasons the central claim is conditional on an unverified regularity premise. I do not recommend rejection: the theorem appears correct conditional on Assumption 3.5, but the paper should either provide a certified supremum check on real trained networks or prove a relaxation of the sup_x condition.","tokens_in":71250,"tokens_out":20377,"duration_ms":231647,"concrete_test":"Retrain the MNIST model with the stated configuration (lucidrains/denoising-diffusion-pytorch, linear β from 10^-4 to 0.02, T=1000) and compute certified upper bounds on max_{x∈R^d}|∂_l s_t^(j)(x)| and max_{x∈R^d}|∂²_lk s_t^(j)(x)| at t=0.99T, using a layer-wise Lipschitz/curvature tool such as α-CROWN or interval bound propagation over an expanding ℓ∞ box, combined with a tail argument using the linear-growth bound ||s_t(x)||≤τ^{-1}fW(√d+||x||). If the certified supremum is infinite or exceeds eK τ^{-2} and eK τ^{-3}, Assumption 3.5 fails for that trained network; if finite, report the resulting eK to check the theorem's constants and step-size condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Assumption 3.5 is the load-bearing premise: Proposition 5.2 and its consequences (Lemmas B.2–B.5, C.8–C.10, and Lemma 5.6) all use the sup_x bounds on |∂_l s_t^(j)| and |∂²_lk s_t^(j)| to control Jacobians, second derivatives, and invertibility of the Runge–Kutta maps. If a trained score network has derivative growth in ||x|| or uncontrolled local curvature, the constants eL^(1)=1+d eK τ^{-2}, eK^(2)=eK τ^{-3} are not meaningful and conditions (45), (48), and the H_max condition in Theorem 3.7 cannot be certified. Section 6.1 states that it evaluates 'sup_{x∈R^d}' but reports only per-pixel maximum images; it does not describe the region searched, the discretization used, or any tail or certified bound. Inspecting maxima over samples or a fixed grid cannot establish a supremum over R^d. Remark 3.6 relaxes only the second-derivative assumption to an L¹(ϱ) condition, not the first-derivative sup, so the issue persists. The central theorem therefore rests on an empirical premise that the numerical section does not actually verify.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper analyzes deterministic samplers for diffusion models based on probability-flow ODEs discretized with p-th order (exponential) Runge–Kutta methods. Under Assumptions 3.1–3.5 it proves a total-variation bound (Theorem 3.7) of the form TV(ϱ_{T−τ}, eϱ_{T−τ}) ≤ TV(ϱ_0, eϱ_0) + C_score τ^{-2} T^{3/4} d^{7/4} ε_score^{1/2} + C_RK T H_max^p d^{p+1} τ^{-γ1}(D + √log d)^{γ2}, with an extension to general forward processes with arbitrary variance schedules in Theorem 4.2. The proof decomposes the one-step error into an initial error, a score-matching error, and a discretization error (Propositions 5.3 and 5.4), and develops detailed estimates for the relevant flow maps and densities in the appendices. Numerical experiments on Gaussian mixtures demonstrate the expected order in H_max and a linear dependence on ε_score, while experiments on MNIST and FashionMNIST are presented as verification of the derivative bounds in Assumption 3.5.","tokens_in":71294,"tokens_out":9750,"duration_ms":113840,"significance":"If the results are correct, they provide polynomial-in-dimension total-variation guarantees for high-order deterministic samplers while requiring only C2 regularity of the learned score, and they cover both standard and exponential Runge–Kutta schemes under non-uniform time steps. The paper has clear strengths: an explicit error decomposition, a structured proof with detailed appendices, concrete Butcher tableaus, public code for the experiments, and falsifiable convergence predictions that are tested numerically. The main weakness is that the key regularity assumption, Assumption 3.5, is presented as 'verified in practice' by Section 6.1, but that section does not actually compute the claimed supremum over R^d. Since the global sup-norm bounds feed directly into Proposition 5.2 and all subsequent estimates, the practical relevance of the theorem depends on this verification, and as it stands the claim of numerical confirmation is overstated.","major_comments":[{"comment":"The section claims to evaluate sup_{x∈R^d} max_{l,j} |∂_l s_t^{(j)}(x)| and sup_{x∈R^d} max_{l,k,j} |∂²_{lk} s_t^{(j)}(x)|, but it only reports per-pixel maximum images. No search region, discretization, optimization procedure, or tail/certified bound is described. Taking maxima over a finite set of samples or a fixed grid cannot establish a supremum over R^d, and a trained network may behave differently far from the data manifold. Because Proposition 5.2 and the estimates that drive the proof (Lemmas B.2–B.5, C.8–C.10, and Lemma 5.6) rely on the global sup norms in Assumption 3.5, the claim that this assumption is verified in practice is not supported. The authors should either provide a transparent method for estimating the sup (for example, an exhaustive scan over a bounded box together with a rigorous tail bound, including a discussion of the network's behavior at infinity) or revise the text so that the experiments are described as suggesting boundedness on the empirical support rather than verifying Assumption 3.5.","section":"Section 6.1"},{"comment":"The theorem is conditional on the uniform bounds sup_x |∂_l s_t^{(j)}(x)| ≤ eK τ^{-2} and sup_x |∂²_{lk} s_t^{(j)}(x)| ≤ eK τ^{-3} for all x ∈ R^d. These bounds are load-bearing: Proposition 5.2 uses them to control the vector fields and Runge–Kutta maps, and the invertibility requirements in Lemma 5.6 and the estimates in the appendices depend on them. Many practical score networks can have derivative growth with ||x|| or uncontrolled local curvature off the data manifold, so this assumption is not automatically satisfied. The paper does not provide a mathematical argument or reference showing that trained diffusion networks satisfy such global uniform derivative bounds. My recommendation is to treat Assumption 3.5 explicitly as an idealization whose practical validity is unknown outside the training distribution, and either strengthen the numerical verification or temper the abstract's claim that the experiments 'further confirm that the derivatives of the learned score function are bounded in practice.'","section":"Assumption 3.5 / Theorem 3.7"},{"comment":"The remark claims that the pointwise second-derivative assumption in Assumption 3.5 can be replaced by the L¹(ϱ_t) condition max_{l,k,j} ∫ |∂²_{lk} s_t^{(j)}(x)| ϱ_t(x) dx ≤ eK τ^{-3}. However, the proof as written repeatedly uses sup_x bounds on ∇² bV and ∇² φ_H (see, e.g., Lemmas B.2, B.5, C.8, and D.2). An L¹ formulation cannot directly replace those sup-norm estimates without additional arguments. The remark is therefore misleading unless a proof of the claimed relaxation is supplied, or it is deleted and replaced by a statement that the L¹ version is an open question.","section":"Remark 3.6"}],"minor_comments":[{"comment":"In the definition of I3, the integrand uses the same symbol ϱt for the density before and after applying the numerical flow; one of the two occurrences should be a different symbol, such as ϱ̃t, to avoid confusion.","section":"Section 5.1, Eq. (42)"},{"comment":"The set of Runge–Kutta evaluation times is written as '{t + c1H, t+ c1H, · · ·, t+ csH}', with a duplicated first entry; it should read '{t + c1H, t + c2H, · · ·, t + csH}'.","section":"Section 5.1, after Eq. (28)"},{"comment":"In the displayed bound, the term '∥ bV_{t+c_sH}(ψ_{s,H}(x)) − V_{t+c_iH}(ψ_{i,H}(x))∥_2' mixes indices c_s and c_i; the two terms should use the same stage index.","section":"Lemma C.9"},{"comment":"The Gaussian-mixture target does not satisfy Assumption 3.1 (compact support), and the artificial score error has discontinuous second derivative, so the convergence experiments are heuristic tests rather than exact settings of the theorem; the text should state this limitation more prominently when interpreting the empirical rates.","section":"Section 6.2"},{"comment":"The manuscript uses many tildes, hats, and barred symbols (ϱt, eϱt, bϱt, fMt, M t, ϕh, eϕh) that look similar in print; a notation table or a short style guide would improve readability.","section":"Notation"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a serious technical contribution and the main theorem appears to be defensible under its stated assumptions. The central issue is that the advertised numerical verification of Assumption 3.5 does not actually compute the claimed supremum over R^d, and this assumption is load-bearing for the entire proof. This is fixable: the authors could provide a more careful empirical procedure or substantially soften the claim that the assumption is verified in practice. I recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is the first non-asymptotic total-variation guarantee for p-th order (exponential) Runge-Kutta solvers on probability-flow ODEs that assumes only C² boundedness of the learned score, allows non-uniform step sizes, and covers general variance schedules. The Gagliardo-Nirenberg step that drops the smoothness requirement from C^{p+1} to C² is the real technical contribution, and the proof is structured and readable: the single-step bound decomposes into the initial error I1, score error I2, and discretization error I3, with propositions doing the heavy lifting and the appendices filling in. Theorem 3.7 and Theorem 4.2 are the product. There is no circularity: scheme coefficients come from order conditions, not from matching the target bound.\n\nThe soft spots, in proportion. Assumption 3.5 is load-bearing: every estimate on vector fields and Runge-Kutta maps uses uniform sup_x bounds on the first and second derivatives of the learned score. Section 6.1 claims to verify sup over R^d but actually shows per-pixel maxima over samples; that cannot certify a supremum over all of R^d, and trained networks can have derivative growth in ||x||. This is the biggest gap between the theorem and the numerical story. The fix is a real verification protocol with certified bounds, or an explicit statement that the theorem is conditional on a premise the experiments make plausible but do not prove.\n\nTheorem 4.2 is sketched, with Lemma 5.7 compressed in A.2; acceptable for a first pass, but should be expanded before publication. The abstract says 'arbitrary variance schedules', while the theorem requires Assumption 4.1 and the step restriction H_max ≤ (d log 2d)^{-1} τ^3 Δ_disc; Remark 3.11 is honest about the restriction, but the abstract oversells. The bound is non-sharp by the authors' own admission—experiments show linear dependence on ε_score and no dimension dependence—and they state this plainly, which is to their credit.\n\nThe math appears coherent. The analysis leans on the authors' prior [28], but the extension is substantive, and citing their own framework is legitimate. I did not find a load-bearing flaw in the proof structure. What keeps me from full confidence is the heuristic verification of Assumption 3.5 and the absence of machine-checked proof; neither is disqualifying.\n\nThis paper is for anyone working on theory of deterministic diffusion samplers. It deserves a serious referee. Recommendation: send to peer review, requiring (1) an honest fix to the abstract's scope claim, and (2) either a stronger numerical verification of Assumption 3.5 or an explicit conditional framing of the main theorems.","headline":"A real advance in deterministic diffusion sampling theory—C² score smoothness suffices—but the empirical verification of the load-bearing smoothness assumption is not yet rigorous.","tokens_in":72082,"tokens_out":3358,"would_cite":true,"duration_ms":36258,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65L06","68T07","60J60"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that deterministic samplers for diffusion probabilistic models—probability flow ODEs solved with p-th order (exponential) Runge-Kutta schemes—converge in total variation with polynomial dependence on dimension, provided…","keywords":["score-based generative models","probability flow ODE","Runge-Kutta methods","exponential integrators","total variation convergence","diffusion probabilistic models","score matching"],"falsifier":"Use a score network that is not twice differentiable—for instance a ReLU network, which fails Assumption 3.5 because its second derivative does not exist as a bounded function—or compute the actual supremum of $|\\partial^2_{lk}s_t^{(j)}(x)|$ on expanding spheres away from the data support; the first test would show that the stated regularity assumption excludes a standard trained model, and the second would settle whether the bound has any footing for the models actually used.","tokens_in":70822,"feed_emoji":"⚙️","tokens_out":15073,"duration_ms":151747,"temperature":0.7,"pith_summary":"Diffusion models generate data by learning the score of a noise-corrupted process and then reversing that process; the fastest practical samplers treat the reversal as a deterministic probability flow ODE and integrate it with a handful of high-order Runge-Kutta steps. This paper sets out to prove that such high-order deterministic samplers converge, showing that the total variation distance between the sampled distribution and the true target is bounded by $O(d^{7/4}\\varepsilon_{\\mathrm{score}}^{1/2} + d(dH_{\\max})^p)$ when the $L^2$ score error is $\\varepsilon_{\\mathrm{score}}$, the dimension is $d$, and the largest solver step is $H_{\\max}$. The required regularity is mild compared with prior analyses: only the first and second derivatives of the learned score need to be bounded, and the result covers non-uniform step sizes and general forward processes with arbitrary variance schedules through enhanced exponential Runge-Kutta schemes. If the proof holds, the common practice of sampling in roughly ten to twenty score evaluations is on a polynomial-convergence footing, with error that scales polynomially in dimension rather than exponentially in the score's Lipschitz constant. The authors also check numerically on image data that the first and second derivatives of trained scores stay bounded, as the assumption requires.","feed_headline":"Bounded score curvature suffices for fast diffusion sampling","feed_subtitle":"Total-variation error stays polynomial: score error to the half power plus d(dH)^p.","key_machinery":"The argument is carried by two discrete flows over a single step: the map $\\phi_H$ built from the true score with the Runge-Kutta coefficients, the map $\\tilde{\\phi}_H$ built from the estimated score, and the interpolation $\\varphi_s$ between their inverses. The total variation error is split as $I_1 + I_2 + I_3$, where $I_1$ is the initial distribution error, $I_2$ is the score-matching error, and $I_3$ is the discretization error. $I_2$ is controlled by pushing the reference density through the interpolation and applying the Gagliardo-Nirenberg inequality to trade $L^1$ quantities for the $L^2$ score error, producing the $d^{7/4}\\varepsilon_{\\mathrm{score}}^{1/2}$ term; $I_3$ is controlled by rewriting the numerical update as an ODE flow and comparing it with the exact flow, producing the $H^p$ term. The load-bearing objects are Assumption 3.5, the uniform derivative bounds $\\sup_x|\\partial_l s_t^{(j)}(x)| \\le \\tilde{K}\\tau^{-2}$ and $\\sup_x|\\partial^2_{lk}s_t^{(j)}(x)| \\le \\tilde{K}\\tau^{-3}$, and Lemma 5.6, which establishes that the Runge-Kutta maps are diffeomorphisms and forces the step-size restriction $H_{\\max} \\le (d\\log 2d)^{-1}\\tau^3\\Delta_{\\mathrm{disc}}$. For general forward processes, the enhanced exponential Runge-Kutta schemes of Section 4.2 rewrite the dynamics so that the same propositions apply to the rescaled score.","core_discovery":"The central claim is Theorem 3.7, extended to general variance schedules as Theorem 4.2: if the target distribution has compact support, the discrete-time weighted $L^2$ score matching error is at most $\\varepsilon_{\\mathrm{score}}^2$, all Runge-Kutta evaluation times lie on the score-matching grid, and the learned score has uniformly bounded first and second derivatives (pointwise bounds $\\tilde{K}\\tau^{-2}$ and $\\tilde{K}\\tau^{-3}$), then the numerical sampler obeys $TV(\\varrho_{T-\\tau}, \\tilde{\\varrho}_{T-\\tau}) \\le TV(\\varrho_0, \\tilde{\\varrho}_0) + C_{\\mathrm{score}}\\tau^{-2}T^{3/4}d^{7/4}\\varepsilon_{\\mathrm{score}}^{1/2} + C_{\\mathrm{RK}} T H_{\\max}^{p}d^{p+1}\\tau^{-\\gamma_1}(D+\\sqrt{\\log d})^{\\gamma_2}$, provided $H_{\\max} \\le (d\\log 2d)^{-1}\\tau^3\\Delta_{\\mathrm{disc}}$. Treating the rollout time $T$, the stopping margin $\\tau$, and the data diameter $D$ as constants, this is the advertised $O(d^{7/4}\\varepsilon_{\\mathrm{score}}^{1/2} + d(dH_{\\max})^p)$, which yields an iteration complexity of $O(d^{1+1/p}\\varepsilon^{-1/p})$ when the score error is negligible. Inside the bound, the score-error term carries the $\\varepsilon^{1/2}$ exponent produced by the Gagliardo-Nirenberg interpolation step, and the discretization term carries $p$-th order convergence in the step size; the two errors add rather than compound one another. The authors state the result the way a numerical analyst would: sampling difficulty is set by the score approximation error and the solver order, not by an exponential contraction estimate of the dynamics.","pith_inferences":["The twice-differentiability assumption excludes the cheapest score architectures, since networks with piecewise-linear activations are not $C^2$; a natural stress test is whether the observed convergence rates survive such networks, which would indicate the regularity assumption can be relaxed to an averaged or distributional condition.","The numerical observation that total variation error scales linearly with $\\varepsilon_{\\mathrm{score}}$ on Gaussian mixtures suggests the $\\varepsilon^{1/2}$ exponent is an artifact of the interpolation step rather than an intrinsic rate; tightening it would likely require an $L^1$-to-$L^2$ stability estimate for the transported density.","The step-size restriction $H_{\\max} \\le (d\\log 2d)^{-1}\\tau^3\\Delta_{\\mathrm{disc}}$ is demanded only to keep the Runge-Kutta maps invertible, yet the experiments show RK4 working with as few as 8 steps; separating the invertibility condition from the convergence proof could lead to dimension-independent step-size rules.","The worst-case dimension factors $d^{7/4}$ and $d^{p+1}$ come from moment bounds over the full ambient space, while the reported dimension-independence in the mixture tests hints that an intrinsic-dimension version of the argument could replace $d$ by the effective dimension of the data support."],"forward_implications":["Any $p$-th order (exponential) Runge-Kutta sampler for a probability flow ODE converges in total variation with error $O(d^{7/4}\\varepsilon_{\\mathrm{score}}^{1/2} + d(dH_{\\max})^p)$, so reaching accuracy $\\varepsilon$ needs $O(d^{1+1/p}\\varepsilon^{-1/p})$ steps when the score error is negligible.","Only boundedness of the first two derivatives of the estimated score is required, not control of the difference between true and estimated score derivatives as in earlier analyses, and not the higher-order smoothness the authors' own prior work demanded.","The guarantee covers non-uniform time grids and general forward processes with arbitrary variance schedules, through the enhanced exponential Runge-Kutta schemes whose evaluation times align with the score-matching grid.","The score error and the discretization error add rather than amplify each other, so improving one source of error does not destabilize the other, and with standard-normal initialization the initial-condition error decays as $e^{-T}\\sqrt{d}D$."],"supporting_citations":[{"why":"The authors' earlier analysis of the probability flow ODE with p-th order Runge-Kutta solvers, which required boundedness of the first p+1 score derivatives and supplied derivative and moment estimates reused here; the present paper relaxes the smoothness requirement to two derivatives.","marker":"[28]"},{"why":"The exponential-Runge-Kutta based deterministic solver family whose schemes the enhanced exponential integrators of Section 4.2 extend, adjusting the evaluation-time alignment with the discrete score grid.","marker":"[48]"},{"why":"Provides the theory of exponential Runge-Kutta integrators and their order conditions, which underwrites the second- and third-order schemes constructed for the general variance-schedule setting.","marker":"[27]"},{"why":"A recent sharp convergence theory for the probability flow ODE requiring control of the difference between true and estimated score derivatives; Table 1 compares the present theorem's assumptions and iteration complexity against it.","marker":"[41]"},{"why":"An earlier probability flow ODE convergence analysis with polynomial dimension dependence that also assumes derivative-difference control, serving as a baseline whose assumptions the present work weakens.","marker":"[40]"},{"why":"Introduces the probability flow ODE and the score-matching objective that define the sampling problem, so the paper's reverse-process ODE and learned-score setup inherit its formulation.","marker":"[62]"},{"why":"The discrete-time weighted score matching formulation, including the grid T with weights sigma-squared, on which Assumption 3.2 and the grid-alignment requirement of Assumption 3.4 are based.","marker":"[25]"},{"why":"A second-order solver convergence analysis for probability flow ODEs that this paper extends to arbitrary orders p and to exponential Runge-Kutta schemes.","marker":"[39]"}],"fun_headline_variants":["High-order ODE solvers converge fast under bounded score curvature","TV error bound: score half-power plus d(dH)^p for diffusion samplers","Rigorous convergence for RK schemes in diffusion ODE sampling","Score derivative bounds ensure fast high-order diffusion sampling","Convergence analysis: O(d^{7/4} sqrt(eps) + d(dH)^p)"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is Assumption 3.5, that the learned score function has first and second derivatives uniformly bounded across all of $\\mathbb{R}^d$ with the stated $\\tau$-scaling; a trained network whose derivatives grow with distance from the data would void the bound, and the paper's empirical check displays pixel-wise maxima over samples rather than computing genuine suprema over $\\mathbb{R}^d$.","fun_headline_variants_meta":{"raw":{"variants":["High-order ODE solvers converge fast under bounded score curvature","TV error bound: score half-power plus d(dH)^p for diffusion samplers","Rigorous convergence for RK schemes in diffusion ODE sampling","Score derivative bounds ensure fast high-order diffusion sampling","Convergence analysis: O(d^{7/4} sqrt(eps) + d(dH)^p)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001135,"raw_usage":{"total_tokens":4853,"prompt_tokens":1223,"completion_tokens":3630,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":839,"completion_tokens_details":{"reasoning_tokens":3531}},"tokens_in":839,"tokens_out":3630,"duration_ms":31166,"temperature":1.0,"reasoning_tokens":3531,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:39:15.402522+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Use a score network that is not twice differentiable—for instance a ReLU network, which fails Assumption 3.5 because its second derivative does not exist as a bounded function—or compute the actual supremum of $|\\partial^2_{lk}s_t^{(j)}(x)|$ on expanding spheres away from the data support; the first test would show that the stated regularity assumption excludes a standard trained model, and the second would settle whether the bound has any footing for the models actually used.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' earlier analysis of the probability flow ODE with p-th order Runge-Kutta solvers, which required boundedness of the first p+1 score derivatives and supplied derivative and moment estimates reused here; the present paper relaxes the smoothness requirement to two derivatives."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The exponential-Runge-Kutta based deterministic solver family whose schemes the enhanced exponential integrators of Section 4.2 extend, adjusting the evaluation-time alignment with the discrete score grid."}],"review_version":1}