{"id":"fa47b77c-edff-4de5-b9a6-4016db3f9322","arxiv_id":"2608.09494","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Walk-on-spheres Monte Carlo and neural network approximators for elliptic PDEs with drift and killing achieve polynomial-in-dimension complexity, extending the drift-free analyses of Beznea et al. and Grohs-Herrmann.","lead":"This paper proves that a Monte Carlo method for elliptic equations, the walk-on-spheres algorithm, can handle equations with drift and decay, with error bounded uniformly over the domain and costs growing only polynomially with dimension. It also proves matching neural-network approximation bounds, extending a research program that aims to show neural PDE solvers can beat the curse of dimensionality.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 5.6's ANN complexity rests on hypothesis (138), yet no domain is shown to admit distance-function ReLU approximations with an absolute exponent η; standard rates leave this condition unsupported.","rationale":"In good faith, the paper's main Monte Carlo contribution appears mathematically sound: the Girsanov reduction in Proposition 2.2, the occupation-measure representation in Lemma 3.9, the bias estimates in Lemmas 4.1 and 4.2, and the L∞-error assembly in Theorem 4.9 all cohere, with no circular step or fitted parameter. The complexities in Corollary 4.10 are derived, not assumed, and the dimension dependence is polynomial as claimed. The reader's conditional verdict is therefore appropriate for the Monte Carlo half. For the ANN half, however, the decisive condition is hypothesis (138). The proof of Theorem 5.6 is a clean probabilistic-method construction once (138) is granted: the clipped network b_ε is a (1/3, δ_ε)-distance, and the ANN parameter bounds follow from Lemmas 5.3–5.5. But the paper does not demonstrate that any domain actually satisfies (138) with the required absolute exponent η. Because r is only Lipschitz for general convex domains, the classical ReLU approximation theory for d-variate Lipschitz functions would give an ε^{-d} parameter dependence, which is incompatible with an absolute η. The abstract's claim of polynomial-in-d ANN approximation is thus contingent on a condition whose feasibility for natural convex domains is not established. This is precisely the reader's weakest_assumption, so the stress-test agrees. The paper also states Theorem 5.6 only for h1 ≡ 0, with the h1 case sketched in Remark 5.7(iii); while this narrows the abstract's wording, it is a presentation issue rather than a correctness flaw. The absence of numerical experiments is not a mathematical objection. Verdict remains CONDITIONAL: the Monte Carlo result stands, and the ANN result is valid conditionally on (138), but the authors should either cite or construct networks satisfying (138) for a concrete non-polytopal domain, or state explicitly that the dimension-uniform ANN claim depends on an unproved hypothesis.","tokens_in":35596,"tokens_out":16333,"duration_ms":191048,"concrete_test":"Fix D = {x ∈ R^d : ||x|| ≤ 1}, the unit ball, and r(x) = 1 - ||x||. Construct explicit ReLU networks r_ε, ε ∈ (0,1), using known Euclidean-norm approximation lemmas (squaring and sqrt networks), and compute P(r_ε) and the Lipschitz constant as functions of d and ε. Hypothesis (138) holds with an absolute η iff P(r_ε) ≤ C ε^{-η} with η and C independent of d up to polynomial factors; if a lower-bound argument shows P(r_ε) ≥ ε^{-c d} for some c > 0, then no absolute η exists and Theorem 5.6 does not yield dimension-uniform polynomial rates. Run D = [0,1]^d as a sanity check, where r is exactly representable with O(d) ReLU units (η = 0).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central ANN statement, Theorem 5.6, claims P(Ψ_ε) ≤ c ε^{-q} with q an absolute constant only under hypothesis (138): the distance-to-boundary r(x) = dist(x, ∂D) must be uniformly approximated by ReLU networks r_ε with P(r_ε) ≤ C ε^{-η}, η absolute, and a uniform Lipschitz bound C. The paper neither proves nor exemplifies (138); Remark 5.7 only discusses consequences. For a generic convex domain r is merely 1-Lipschitz, and the standard ReLU approximation rate for d-variate Lipschitz functions is O(ε^{-d}) parameters, which would force η to depend on d. If the minimal η for D = unit ball or for a smooth convex body is not bounded independently of d, then P(Ψ_ε) ≤ c ε^{-q} still holds for each fixed d, but the d-dependence is absorbed into c (or q), and the advertised 'overcoming the curse of dimensionality' in Remark 5.7(i) fails. Since the entire ANN half of the abstract rests on this conditional, and no instantiation is given, this is the weakest load-bearing point. The Monte Carlo half (Corollary 4.10) is not affected: it is unconditional given convexity and a (β, δ)-distance.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the numerical approximation of functions w(x) = E[h0(τ^x, x + B_{τ^x})] + E[∫_0^{τ^x} h1(s, x + B_s) ds] on bounded convex domains D, which arise as stochastic representations of solutions of linear elliptic PDEs with constant diffusion, drift, and killing (Eq. (1), Prop. 2.2). The authors extend the modified Walk-on-Spheres framework of Beznea et al. [6] by explicitly including the sampled exit times needed for drift and killing, construct Monte Carlo estimators of the form (5), and prove a uniform L∞ error bound (Thm. 4.9) together with a polynomial complexity statement (Cor. 4.10). They then prove an ANN approximation result (Thm. 5.6) for the case h1 ≡ 0, under the assumptions that the boundary data are representable by an ANN and that the distance-to-boundary function is uniformly approximable by ReLU networks with a certain rate (condition (138)). The paper concludes with remarks on overcoming the curse of dimensionality.","tokens_in":35647,"tokens_out":10018,"duration_ms":125684,"significance":"If the results hold, the Monte Carlo half is a solid contribution: it extends the uniform-error complexity analysis of [6] and [11] to drift-diffusion-reaction equations, with sample complexity polynomial in 1/ε and, under explicit growth assumptions, in the dimension d. The proof chain for the Monte Carlo part is detailed and appears sound, including the Itô-based moment estimates (Lemmas 3.6, 3.7), the bias decomposition (Lemmas 4.1, 4.2), and the randomized grid argument in Theorem 4.9. The ANN half is more delicate: Theorem 5.6 is a conditional statement whose central hypothesis (138) is not instantiated for any concrete domain or class of domains, and as written the proof contains a parameter inconsistency around Eq. (145). The paper is therefore best assessed as a partially conditional extension of known techniques rather than a fully unconditional resolution of the high-dimensional approximation problem.","major_comments":[{"comment":"The main ANN result rests on the assumption that for every ε ∈ (0,1) the function r(x) = dist(x, ∂D) is uniformly approximated by ReLU networks r_ε with P(r_ε) ≤ C ε^{-η} for an absolute constant η and a uniform Lipschitz constant C. The paper neither proves this for any domain D nor cites a class of domains for which it holds. For a generic convex domain, r is only 1-Lipschitz, and the standard ReLU approximation rate for d-variate Lipschitz functions is of order ε^{-d}, which would make η dimension-dependent. In that case the conclusion P(Ψ_ε) ≤ c ε^{-q} still holds for each fixed d, but the dimension dependence is absorbed into c (or q), and the 'overcoming the curse of dimensionality' claim in Remark 5.7(i) is not established. The authors should either prove (138) for explicit non-trivial examples (e.g., Euclidean balls or convex polytopes), or substantially weaken the interpretation of the theorem as a purely conditional result.","section":"§5.3, Theorem 5.6, condition (138) and Remark 5.7"},{"comment":"In the proof of Theorem 5.6, the network b_ε is defined by R(b_ε)(x) = ((R(r_{δ_ε/3}))(x) − ε)_+ and is then claimed to be a (1/3, δ_ε)-distance. With the parameter choice in (140), δ_ε is typically much smaller than ε (frequently δ_ε ≈ ε^2), so for a point x with r(x) ≥ δ_ε the lower bound R(b_ε)(x) ≥ δ_ε/3 is not guaranteed: subtracting ε instead of δ_ε/3 can make the positive part vanish. The intended definition is almost certainly R(b_ε)(x) = ((R(r_{δ_ε/3}))(x) − δ_ε/3)_+, which does combine with the approximation error in (138) to yield a (1/3, δ_ε)-distance. As written, however, the application of Corollary 4.10 at (150) is not justified, so this is a load-bearing point that must be corrected.","section":"§5.3, Eq. (145)"}],"minor_comments":[{"comment":"In the proof of Lemma 4.4(ii), the displayed bound for |τ̂{x,i}_k − τ̂{y,i}_k| contains a repeated index in the term ∥X̂{x,i}_{k−1} − X̂{x,i}_{k−1}∥; the second superscript should be y, not x. This is a harmless typo but should be fixed.","section":"§4.2, Lemma 4.4"},{"comment":"The definition of w in Eq. (32) would be clearer with explicit parentheses, e.g., w(x) = 1 − γ∥x∥²/(d + γr²), since the intended reading is used immediately afterward in the proof.","section":"§3, Lemma 3.6"},{"comment":"The bound (108) is extremely long and difficult to parse. Splitting it into named aggregate terms, as is partially done in the proof, would improve readability without changing the mathematics.","section":"§4.2, Theorem 4.9"},{"comment":"The extension to h1 ≠ 0 is only sketched, while the abstract and introduction highlight the drift-killing class of PDEs more broadly. A precise statement of the h1 ≠ 0 ANN result, even as a corollary, would make the paper's claims easier to verify.","section":"§5.3, Remark 5.7(iii)"},{"comment":"Reference [6] is cited as the arXiv preprint arXiv:2209.01432; if a published version now exists, it should be cited in its final form.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The Monte Carlo part of the paper appears sound and would be a useful contribution to the literature on Walk-on-Spheres methods for elliptic PDEs with drift and killing. My main reservation concerns the ANN theorem: condition (138) is a strong, unverified structural assumption, and the proof contains a parameter error in the definition of b_ε in Eq. (145). Both issues are fixable in a revision, but they affect the advertised ANN contribution rather than mere presentation. I do not see any indication of circularity or misconduct; the δ_ε/3 versus ε issue looks like a typographical slip."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The Monte Carlo half of this paper is solid and worth engaging; the ANN half is a conditional result resting on a hypothesis that is never instantiated, and the abstract oversells what is proved.\n\nWhat is genuinely new: the estimator (5) with explicitly sampled random times, the occupation-measure representation of the integral term (Lemma 3.9), the L-infinity bias and concentration analysis with drift and killing (Lemmas 4.1/4.2, Theorem 4.9), and the polynomial complexity statement Corollary 4.10. I checked the proof skeleton the reader flagged: Lemma 3.6's sub/super-solution bounds, Lemma 3.7's tail estimate, Lemma 4.5's Ito identities, the grid-plus-concentration cancellation in Theorem 4.9, and the scalings delta ~ eps^2, M ~ eps^{-4} log(1/eps), n ~ eps^{-14}. Everything reconciles. No fitted parameters, no circularity. The Monte Carlo result is a genuine extension of [6] and [11] to drift and killing, unconditional given convexity and a (beta, delta)-distance, and the attribution to the earlier framework is honest.\n\nSoft spots, in proportion. First, Theorem 5.6 proves the ANN bound only for h1 = 0; the integral case is a sketch in Remark 5.7(iii), yet the abstract claims the full representation (1). Presentation gap: prove the extension or narrow the abstract. Second, hypothesis (138), ANN-approximable distance-to-boundary with size C eps^{-eta} for absolute eta, is never exemplified. The stress-test's specific worry about the unit ball seems overstated - the Euclidean norm is not a generic Lipschitz function and has structured ReLU approximations with size polynomial in d and polylog(1/eps), so (138) plausibly holds for balls and smooth convex bodies - but the paper proves none of this. As written, the ANN half's curse-of-dimensionality claim has no worked example behind its load-bearing assumption. One lemma instantiating (138) would fix it. Third, convexity replaces [6]'s exterior ball condition; a restriction, not an error, and the authors are upfront. Fourth, the occupation measure mu in Lemma 3.8 is defined abstractly, with no sampling procedure; the integral-term estimator's practical cost depends on this and is untested. No numerical experiments, which is tolerable for a pure complexity paper.\n\nWho this is for: researchers working on Monte Carlo and neural-network complexity theory for elliptic PDEs, especially the [6]/[11] lineage. I would cite the Monte Carlo half; I would not yet lean on the ANN half. Send it to peer review - a serious referee can verify the MC chain quickly and should push the authors to instantiate (138).","headline":"The Monte Carlo half is solid and genuinely extends the Beznea-Grohs framework to drift and killing; the ANN half is a conditional bound on an uninstantiated distance-function hypothesis, and the abstract oversells the ANN claim.","tokens_in":36431,"tokens_out":7838,"would_cite":true,"duration_ms":78472,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["65C05","65N75","35J15","68T07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A modified Walk-on-Spheres estimator with sampled random times approximates elliptic PDEs with drift and killing uniformly, with Monte Carlo and ReLU-network cost polynomial in accuracy and dimension.","keywords":["Walk-on-Spheres","elliptic PDE with drift and killing","Monte Carlo estimator","deep neural network approximation","ReLU networks","curse of dimensionality","Feynman-Kac representation","uniform error bounds"],"falsifier":"Find a bounded convex domain $D$ and a ReLU-representable boundary datum $h$ for which $\\mathrm{dist}(\\cdot,\\partial D)$ cannot be uniformly $\\varepsilon$-approximated by ReLU networks of size $O(\\varepsilon^{-\\eta})$ for any absolute constant $\\eta$; then the parameter bound of Theorem 5.6 fails for that domain. At the Monte Carlo level, compute $\\mathbb{E}[\\tau^x]$ and the $\\delta^{1/2}$ bias term of Lemma 4.1 on a long thin rectangle to test whether the convexity bound $\\mathbb{E}[\\tau^x]\\le r(x)\\operatorname{diam}(D)$ is the right mechanism.","tokens_in":35092,"feed_emoji":"🎲","tokens_out":11091,"duration_ms":106015,"temperature":0.7,"pith_summary":"The paper proves that a modified Walk-on-Spheres Monte Carlo estimator, and a ReLU neural network built from it, approximate the solution of linear elliptic PDEs with constant diffusion, drift, and killing uniformly over a bounded convex domain. The estimator adds sampled random times to the classical scheme, and those times are what compensate the drift and killing terms in the Feynman-Kac representation. The main error bound says that for a prescribed accuracy $\\varepsilon$ the number of Monte Carlo paths and steps grows at most polynomially in $1/\\varepsilon$ and the dimension $d$. The neural-network theorem proves the boundary-payoff part $w(x)=\\mathbb{E}[h(\\tau^x,x+B_{\\tau^x})]$ is ReLU-approximable with parameter count polynomial in $1/\\varepsilon$ and $d$, assuming the boundary data and distance-to-boundary are network-approximable, with the source-term case noted as an extension. If correct, this moves a broader class of high-dimensional elliptic boundary value problems out of the curse-of-dimensionality regime.","feed_headline":"Walk-on-Spheres Monte Carlo now covers drift-and-killing PDEs","feed_subtitle":"A modified estimator reaches ε-accuracy with work polynomial in 1/ε and dimension; ReLU nets inherit the bound.","key_machinery":"The load-bearing object is the modified Walk-on-Spheres process driven by a $(\\beta,\\delta)$-distance: a Lipschitz function $\\bar{r}$ that stays below the true boundary distance $r$ and keeps at least a $\\beta$ fraction of it until the process is within $\\delta$ of the boundary. At each step the process jumps uniformly on the sphere of radius $\\bar{r}(\\bar{X})$ around the current point, and the physical time $\\bar{\\tau}$ accumulates $\\bar{r}(\\bar{X})^2$ times independent copies of the exit time $\\zeta$ of Brownian motion from the unit ball. This random time is what absorbs drift and killing in the representation. The companion mechanism is the time-space occupation measure $\\mu$ of Lemma 3.8, which converts integrals over Brownian paths inside the unit ball into expectations of samples $(V_k,Y_k)$, so the estimator (5) becomes a plain average of i.i.d. path functionals. The analysis rests on three estimates: a geometric tail bound on the number of steps needed to come within $\\delta$ of the boundary, a bias bound with terms of order $\\delta$ and $\\delta^{1/2}$, and a concentration bound that controls the supremum over the domain through a grid whose size is polynomial in $M$ and the Lipschitz constant of $\\bar{r}$.","core_discovery":"On the paper's own terms, the central discovery is that the stochastic representation $w(x)=\\mathbb{E}[h_0(\\tau^x,x+B_{\\tau^x})]+\\mathbb{E}[\\int_0^{\\tau^x} h_1(s,x+B_s)\\,ds]$ of solutions to (2) can be evaluated by the Walk-on-Spheres estimator (5), in which each step's physical time is accumulated as $\\bar{r}(\\bar{X})^2$ times the exit time of Brownian motion from the unit sphere. For bounded Lipschitz $h_0,h_1$ and a $(\\beta,\\delta)$-distance $\\bar{r}$, Corollary 4.10 gives $\\mathbb{E}[\\sup_{x\\in D}|w(x)-\\bar{w}(x)|]\\le\\varepsilon$ with $\\max\\{n,M\\}\\le c\\,d^q\\varepsilon^{-q}$, where $q$ is an absolute constant and $c$ depends at most polynomially on the problem data. Theorem 5.6 shows that when $h_0$ is representable by an ANN and the boundary-distance function is uniformly approximated by ReLU networks satisfying (138), there exists a ReLU network $\\Psi_\\varepsilon$ whose realization $R(\\Psi_\\varepsilon)$ satisfies $\\sup_{x\\in D}|w(x)-R(\\Psi_\\varepsilon)(x)|\\le\\varepsilon$ with $P(\\Psi_\\varepsilon)\\le c\\varepsilon^{-q}$. The link to the PDE is a Girsanov-based Feynman-Kac formula, after which the multiplier $e^{-\\langle b,x\\rangle}$ converts $w$ into the solution of (2).","pith_inferences":["The neural-network theorem does not exhibit a concrete domain where hypothesis (138) holds; constructing explicit ReLU approximations of $\\mathrm{dist}(\\cdot,\\partial D)$ for high-dimensional balls, cubes, or polytopes and measuring the actual exponent $\\eta$ would determine how widely the polynomial parameter bound applies.","The Monte Carlo half assumes convexity of $D$ to get $\\mathbb{E}[\\tau^x]\\le r(x)\\operatorname{diam}(D)$; if a similar exit-time bound holds under the weaker exterior ball condition used in the drift-free predecessor, the method would extend to nonconvex domains.","The use of sampled random times suggests a general recipe for other Feynman-Kac-type estimators: any additive functional of Brownian exit can be turned into an i.i.d. path average by sampling the time-space occupation of the unit ball, not just the exit position."],"forward_implications":["On any bounded convex domain with bounded Lipschitz data, the drift-killing elliptic PDE (2) can be approximated to uniform accuracy $\\varepsilon$ by Monte Carlo with sample and step counts polynomial in $\\varepsilon^{-1}$ and $d$.","Whenever the boundary data and the distance-to-boundary are ReLU-representable with polynomial-size networks, the solution's boundary-payoff part is ReLU-approximable with polynomial parameter count, breaking the curse of dimensionality on this class.","The same estimator structure covers the source term $h_1\\neq 0$ in the Monte Carlo half, and the authors indicate the neural construction extends to that case as well.","The deterministic $M$-step stopping rule gives the approximating network a fixed depth for each target accuracy, which is what makes the ANN construction feasible."],"supporting_citations":[{"why":"supplies the modified Walk-on-Spheres process with $(\\beta,\\delta)$-distances and deterministic stopping that this paper extends to drift and killing.","marker":"[6]"},{"why":"provides the convex-domain exit-time bound $\\mathbb{E}[\\tau^x]\\le r(x)\\operatorname{diam}(D)$ and the earlier Monte Carlo plus ANN strategy for the Poisson problem.","marker":"[11]"},{"why":"introduces the classical Walk-on-Spheres algorithm whose sphere-exit sampling underlies every estimator in the paper.","marker":"[23]"},{"why":"supplies the facts $\\mathbb{E}[\\zeta]=1/d$ and the Brownian time-space occupation measure used in Lemma 3.8 and Lemma 4.5.","marker":"[28]"},{"why":"provides the ReLU product-network approximation used in Lemma 5.2 to build the time-accumulating networks $\\Theta^k_\\varepsilon$.","marker":"[35]"},{"why":"supplies the ANN composition, parallelization, and summation calculus used to assemble the networks in Lemmas 5.3 through 5.5.","marker":"[13]"},{"why":"gives the parameter-bound lemma invoked to convert the uniform approximation of the Monte Carlo estimator into the final $P(\\Psi_\\varepsilon)\\le c\\varepsilon^{-q}$ statement.","marker":"[16]"},{"why":"shows a prior extension of Walk-on-Spheres to drift-diffusion equations, the baseline the random-time estimator improves upon.","marker":"[30]"}],"fun_headline_variants":["Walk-on-Spheres Monte Carlo now handles drift and killing PDEs","Fast Monte Carlo and ReLU nets for elliptic PDEs with drift and killing","Epsilon-accurate Monte Carlo and DNNs for drift-killing PDEs","Walk-on-Spheres extended to drift and killing with ReLU nets"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The neural-network half stands on the unverified hypothesis that the distance to the boundary can be uniformly approximated to accuracy $\\varepsilon$ by ReLU networks of size $C\\varepsilon^{-\\eta}$ with an absolute constant $\\eta$; the Monte Carlo half separately requires $D$ to be convex, so nonconvex exterior-ball domains are out of reach.","fun_headline_variants_meta":{"raw":{"variants":["Walk-on-Spheres Monte Carlo now handles drift and killing PDEs","Fast Monte Carlo and ReLU nets for elliptic PDEs with drift and killing","Epsilon-accurate Monte Carlo and DNNs for drift-killing PDEs","Walk-on-Spheres extended to drift and killing with ReLU nets"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000904,"raw_usage":{"total_tokens":3948,"prompt_tokens":1065,"completion_tokens":2883,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":681,"completion_tokens_details":{"reasoning_tokens":2800}},"tokens_in":681,"tokens_out":2883,"duration_ms":19823,"temperature":1.0,"reasoning_tokens":2800,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T16:27:20.020201+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a bounded convex domain $D$ and a ReLU-representable boundary datum $h$ for which $\\mathrm{dist}(\\cdot,\\partial D)$ cannot be uniformly $\\varepsilon$-approximated by ReLU networks of size $O(\\varepsilon^{-\\eta})$ for any absolute constant $\\eta$; then the parameter bound of Theorem 5.6 fails for that domain. At the Monte Carlo level, compute $\\mathbb{E}[\\tau^x]$ and the $\\delta^{1/2}$ bias term of Lemma 4.1 on a long thin rectangle to test whether the convexity bound $\\mathbb{E}[\\tau^x]\\le r(x)\\operatorname{diam}(D)$ is the right mechanism.","supporting_citations":[{"cited_title":"Deep neural network approximation for high-dimensional elliptic PDEs with boundary conditions","cited_arxiv_id":null,"evidence_quote":"provides the convex-domain exit-time bound $\\mathbb{E}[\\tau^x]\\le r(x)\\operatorname{diam}(D)$ and the earlier Monte Carlo plus ANN strategy for the Poisson problem."},{"cited_title":"Some continuous Monte Carlo methods for the Dirichlet problem","cited_arxiv_id":null,"evidence_quote":"introduces the classical Walk-on-Spheres algorithm whose sphere-exit sampling underlies every estimator in the paper."},{"cited_title":"Port and Charles J","cited_arxiv_id":null,"evidence_quote":"supplies the facts $\\mathbb{E}[\\zeta]=1/d$ and the Brownian time-space occupation measure used in Lemma 3.8 and Lemma 4.5."},{"cited_title":"Error bounds for approximations with deep ReLU networks.Neural Networks, 94:103–114, 2017","cited_arxiv_id":null,"evidence_quote":"provides the ReLU product-network approximation used in Lemma 5.2 to build the time-accumulating networks $\\Theta^k_\\varepsilon$."},{"cited_title":"Space- time error estimates for deep neural network approximations for differential equa- tions","cited_arxiv_id":null,"evidence_quote":"supplies the ANN composition, parallelization, and summation calculus used to assemble the networks in Lemmas 5.3 through 5.5."},{"cited_title":"Deep neural networks can provably solve Bellman equations for Markov decision processes without the curse of dimensionality","cited_arxiv_id":"2506.22851","evidence_quote":"gives the parameter-bound lemma invoked to convert the uniform approximation of the Monte Carlo estimator into the final $P(\\Psi_\\varepsilon)\\le c\\varepsilon^{-q}$ statement."},{"cited_title":"Sabelfeld","cited_arxiv_id":null,"evidence_quote":"shows a prior extension of Walk-on-Spheres to drift-diffusion equations, the baseline the random-time estimator improves upon."}],"review_version":1}