{"id":"7698d517-c2f8-4cf7-8cc6-959084312eaf","arxiv_id":"2511.16178","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"For a 3D active Brownian particle, the most probable transition path changes from in-plane straight/U-shaped curves to helical 3D paths as final time and displacement are varied.","lead":"A neural network trained to minimize the Onsager-Machlup integral finds the most probable paths of a 3D active Brownian particle. The paths switch from straight or U-shaped in-plane curves to 3D helices as travel time and net displacement change.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"H-path phase diagram rests entirely on NN minimization; no independent checkpoint (EL residual, seed variance, alternative optimizer) shows the reported helical paths are global OMI minima.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing concern: the H-path branch is not independently verified as a global minimum of the OMI. I agree with that assessment. The paper has real strengths—the OMI construction from a Rayleighian with active power is a reasonable extension of established Onsager-Machlup theory, and the 2D I/U validation against Ref. [6] provides meaningful support for the optimizer on those branches. The failure is specific: the new qualitative result (helical H-paths and the phase boundaries in Figs. 2(f) and 3(f)) rests solely on NN optimization with no code, no seed variance, no residual check, and no comparison with an alternative method. This is not an allegation of error but a missing evidential link. A multi-start optimization plus an Euler-Lagrange residual check would settle whether the H-path is a genuine global minimum or a local-minimum artifact. Since the concern is real but addressable, the reader's CONDITIONAL verdict is appropriate and unchanged.","tokens_in":10359,"tokens_out":8419,"duration_ms":88974,"concrete_test":"Run a multi-start optimization for tf=14, xf=5, θ_f=π/2: train the same NN architecture from at least 50 independent random initializations plus one initialization from the 2D U-path branch, using the same loss Eq. (6), and record the OMI and path geometry of the best run. In parallel, compute the Euler-Lagrange residual (footnote [37]) for the reported H-path. If the best multi-start OMI is lower than the published H-path value, or if the EL residual is non-negligible (>1e-2), then the published H-path is not the global MPP and the phase diagram is unsupported. If the best OMI matches and the residual is small, the local-minimum objection is empirically settled for this parameter point.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim—that the phase diagrams in Figs. 2(f) and 3(f) describe true most probable paths—depends on the trained NNs having found the global minimum of the OMI in Eq. (3). The authors explicitly note (Sec. 'Onsager-Machlup integral') that the Euler-Lagrange equations are only necessary, not sufficient, for a minimum, and they provide no independent verification for the new H-path regime. The I- and U-paths are validated against the 2D analytical results of Ref. [6], which gives confidence in the optimizer for those branches. The H-paths, however, are supported only by the NN output: no Euler-Lagrange residual is computed, no second-variation (Hessian) check is reported, no seed-to-seed variance is shown, and no alternative optimizer or boundary-value solver is compared. Because the OMI is non-convex in the joint position-orientation space (terms involve sinθ, cosθ, and products), Adam can converge to a local minimum whose OMI is above the true global minimum. If that happened for H-region parameters, the classification, the OMI curves in Fig. 2(e), and the phase boundaries would be artifacts. The claim that 'the additional out-of-plane degrees of freedom ... permit trajectories with a lower OMI' presupposes the very global minimality that is unverified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a neural-network method to determine the most probable path (MPP) of a three-dimensional active Brownian particle (ABP) by directly minimizing the Onsager-Machlup integral (OMI). The OMI is constructed from a Rayleighian that includes the active power of the self-propulsion (Eqs. (1)-(3)), and the minimization is performed with fully connected neural networks parameterizing the position and orientation as functions of time (Eq. (6)). For boundary conditions with θ_f = π/2, the reported MPPs change from straight in-plane paths (I) to curved in-plane paths (U) and then to three-dimensional helical paths (H) as the final time increases (Fig. 2). For θ_f = 5π/2, the paper reports H-paths and a later in-plane ℓ-path (Fig. 3). The OMI values for the I- and U-paths agree with the 2D analytical results of Ref. [6], while the H-paths are new. The paper claims that the phase diagrams in Figs. 2(f) and 3(f) describe the true most probable paths of the 3D ABP.","tokens_in":10723,"tokens_out":7005,"duration_ms":69480,"significance":"If the central claim is correct, the work provides a general numerical route to transition paths in higher-dimensional active systems without solving nonlinear Euler-Lagrange equations. The derivation of the OMI from the Rayleighian is algebraically clean and the cancellation of the Péclet number from the argmin is a useful simplification. The validation against the 2D analytical results for the I- and U-paths is a genuine strength. The discovery of helical most probable paths, if confirmed to be global minima, would be an interesting and nontrivial result with potential implications for optimal transport and rare-event dynamics in active matter. However, the H-path regime currently rests entirely on the neural-network optimization, and the absence of an independent check of global minimality leaves the central phase diagram claim insufficiently supported.","major_comments":[{"comment":"The phase diagrams for the H-paths are supported only by the NN minimization. The I- and U-paths are cross-validated against the analytical 2D results of Ref. [6], but no independent check is provided for the H-path regime. The paper itself notes that the Euler-Lagrange equations are only necessary conditions (text after Eq. (3)) and that convergence to a minimum is not guaranteed. The OMI is non-convex in the joint position-orientation space, so Adam may settle at a local minimum. Please provide at least one of the following for representative H-path parameters: (i) an Euler-Lagrange residual check, (ii) a second-variation (Jacobi) test, (iii) a seed-to-seed variance analysis, or (iv) a comparison with a different optimizer or a boundary-value solver. Without such a check, the claim that the H-paths are the MPPs, and hence the phase boundaries in Figs. 2(f) and 3(f), is not established.","section":"Minimization of OMI by NN; Figs. 2(e), 2(f), 3(e), 3(f)"},{"comment":"The classification into I-, U-, and H-paths (and ℓ-paths) is described qualitatively: 'For tf ≳9' and 'For tf ≳12' in Fig. 2(d), and 'the torsion decreases to zero' / 'a small annulus appears' in Fig. 3. No quantitative thresholds for average curvature and torsion are stated, nor is the definition of 'H-path' (e.g., a threshold on the circular projection or on torsion) given. Since the phase diagrams in Figs. 2(f) and 3(f) are built on this classification, the criteria must be specified precisely, otherwise the boundaries are not reproducible.","section":"Figs. 2(d) and 3(d)"},{"comment":"The training setup is under-specified. The paper states Adam is used and gives the network architecture and boundary-penalty weights, but it does not report the learning rate, number of training epochs, loss values at convergence, or the number of independent initializations. This is important because the OMI minimization is non-convex and the reported H-paths may depend on the random initialization of the networks. Please provide these details and, if multiple runs were performed, report the variance of the OMI and of the path geometry.","section":"Minimization of OMI by NN"}],"minor_comments":[{"comment":"The text says the NNs output 'position r(t) and orientation e(t)', and Eq. (6) uses K = r, e. However, the boundary conditions are given in terms of angles θ and φ, and the network count is described as 'five NNs'. Please clarify whether the orientation network outputs the Cartesian unit vector e or the angles (θ, φ). If e is output directly, a unit-norm constraint must be imposed; if angles are output, the notation K = e is confusing.","section":"Eq. (6) and Fig. 1(b)"},{"comment":"The same symbol O is used for the dimensional OMI in Eq. (3), for the dimensionless OMI O = 2O/(k_B T Pe), and for the loss term in Eq. (6). Similarly S is used both for the entropy change in Eq. (5) and for its dimensionless version. Please use distinct symbols (e.g., O, Õ, and J_loss) to avoid ambiguity.","section":"Eq. (3) and Eq. (6)"},{"comment":"The black dashed line in Fig. 2(f) and the transition line in Fig. 3(f) are described as 'a guide for the eye'. It would be useful to state how these guides were drawn and to show error bars or confidence regions if the phase boundaries are estimated from noisy OMI comparisons.","section":"Figs. 2(f) and 3(f)"},{"comment":"The paper does not mention the Péclet number value used in the simulations. Since Pe cancels in the minimization, the MPP is independent of Pe, but the validity of the OMI as the path action may require Pe sufficiently large to neglect possible boundary/determinant terms in the discretized path integral. A brief comment on this would be helpful.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper has a solid derivation and a promising method, but the central new claim—the helical H-path phase—needs independent validation to rule out local-minimum artifacts. The authors should be asked to provide the convergence and reproducibility details. If they cannot supply an independent check, the phase diagram claim should be weakened to 'candidate MPPs' or the paper should be limited to the validated I/U regimes."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take on arXiv:2511.16178. The paper does a straightforward, honest thing: it takes the Onsager-Machlup integral for a 3D active Brownian particle, builds it from the Rayleighian with active power, and minimizes it with neural networks. The genuinely new piece is that in 3D you get a helical H-path branch plus boundary-condition-dependent phase diagrams for theta_f=pi/2 and 5pi/2. That is a real extension beyond the 2D analysis in Ref. [6], and the I- and U-path results match the 2D analytical line, which is the right validation.\n\nThe conceptual part is solid. The OMI derivation in Eqs. (1)-(3) is clean; Pe multiplies the whole functional and cancels out of the argmin, so there is no circular fitting. The curvature/torsion order parameters are a sensible way to classify the branches. The observation that changing theta_f by 2pi changes the transition structure is a good one.\n\nThe soft spot is exactly where the reader points: the H-path branch and the phase boundaries that involve it rest entirely on the NN minimization. The authors note themselves that the Euler-Lagrange equations are only necessary, not sufficient, for a minimum. They do not show an Euler-Lagrange residual, a second variation, seed-to-seed variance, or a comparison with a shooting method for the EL equations. Without that, the claim that the out-of-plane degrees of freedom 'permit trajectories with a lower OMI' is plausible but not demonstrated to be the global minimum. I would not call it a fatal flaw, because the 2D validation gives real confidence in the optimizer, and the I/U branches line up. But the H-region phase boundaries in Figs. 2(f) and 3(f) are not yet fully supported.\n\nA smaller point: the active Rayleighian from Ref. [23] is stated rather than defended. That is fine for a Letter, but it should be explicit that the entire path-probability functional depends on that modeling choice. The classification thresholds for curvature/torsion are also not pinned down; minor.\n\nIf I were the editor: send it to peer review. The result is useful to people who compute transition paths in active matter, and the missing convergence evidence is checkable in a revision. I would want the authors to add one independent check for the helical branch, and report seed variance, before I believed the phase diagram completely. It is worth a serious referee's time.","headline":"A clean 3D extension of the 2D ABP most-probable-path analysis that gives genuinely new helical paths, but the new branch lacks an independent global-minimum check.","tokens_in":11219,"tokens_out":2488,"would_cite":true,"duration_ms":21935,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["05.40.-a","05.40.Jc"],"model":"deepseek-v4-flash","headline":"For a free 3D active Brownian particle, the most probable path between fixed boundary states is not always straight: as the final time or net displacement grows, the optimal path becomes an in-plane curve and then a three-dimensional helix.","keywords":["active Brownian particle","Onsager-Machlup integral","most probable path","neural network optimization","Rayleighian","helical paths","Langevin bridge","transition paths"],"falsifier":"Compute the Euler-Lagrange residuals of the optimized helical paths; if the residuals do not vanish along the trajectory, or if retraining from many random initializations converges to different OMI values, then the reported H-path is not the true most probable path.","tokens_in":10290,"feed_emoji":"🌀","tokens_out":6007,"duration_ms":65035,"temperature":0.7,"pith_summary":"The paper tries to establish that the most probable path of a free 3D active Brownian particle can be found by directly minimizing the Onsager-Machlup integral, and that this path changes geometry as the final time and net displacement vary. By including the active self-propulsion power in the Rayleighian, the authors extend the Onsager-Machlup variational principle to an active nonequilibrium system. The result is a concrete phase behavior: straight in-plane paths, curved planar paths, and three-dimensional helical paths, with boundary orientations controlling where the transitions occur. If correct, this gives a general numerical route to transition paths in higher-dimensional active systems without solving the nonlinear Euler-Lagrange equations.","feed_headline":"Most probable path of a 3D active particle turns helical","feed_subtitle":"Minimizing the Onsager-Machlup integral with neural nets reveals helical transition paths as time or displacement grows.","key_machinery":"The central object is the Onsager-Machlup integral (OMI), the time integral of the Rayleighian measured relative to its deterministic minimum. For the 3D ABP it takes the dimensionless form O ∝ ∫[(ẋ−sinθ cosφ)² + (ẏ−sinθ sinφ)² + (ż−cosθ)² + θ̇² + φ̇² sin²θ] dt, with the Péclet number as prefactor. The Rayleighian is constructed by adding the active power ζ_t U·v to the dissipation function, so the Onsager-Machlup principle applies even though the active force has no free energy. The minimization is carried out by fully connected neural networks that parameterize r(t) and e(t), with time derivatives computed by automatic differentiation and boundary conditions enforced as soft constraints","core_discovery":"For a 3D active Brownian particle with prescribed initial and final positions and orientations, the most probable path is obtained by minimizing the dimensionless Onsager-Machlup integral. Direct neural-network minimization reveals that, as the final time increases (or as the apparent velocity x_f/t_f decreases), the most probable path transitions from a straight in-plane I-path, to a curved planar U-path, to a three-dimensional helical H-path that uses the extra transverse dimension to lower the OMI. When the final orientation includes an extra 2π rotation, a competing in-plane ℓ-path appears. The I- and U-path results match the known 2D analytical solutions, while the H-paths are new; the","pith_inferences":["Although the paper focuses on a free particle, the same neural variational framework should extend to ABPs in external potentials or with obstacles, where rare transitions between metastable states become physically relevant.","The reported discontinuity in curvature and torsion at the I→U and U→H transitions hints at a buckling-like bifurcation; if pursued, an analogy with elastica theory might yield analytic estimates for the transition lines.","If helical most probable paths are genuinely the global minima, they should be observable in tracking experiments of self-propelled colloids or microswimmers with prescribed start and end configurations, where the curvature and torsion signatures could be measured directly.","The method could be pushed further by maximizing the modified, shifted OMI rather than minimizing it, giving access to cumulant generating functions and fluctuation statistics beyond the single most probable path."],"forward_implications":["The I→U→H transition implies that apparent velocity x_f/t_f and final time are control parameters for path geometry, allowing phase diagrams for most probable paths in active matter.","The 2D agreement for I- and U-paths shows the neural minimization reproduces known analytical results, so the method can serve as a benchmark for other variational approaches.","Boundary orientations can shift transition lines substantially, meaning experimental preparation of initial and final orientations can select between planar and helical transition paths.","The same active Rayleighian construction with added active power can be applied to other nonequilibrium and nonreciprocal active systems to compute their most probable transition paths.","The OMI and entropy change evaluated along the most probable path provide direct estimates of the dissipation and time-reversal asymmetry of the dominant transition route."],"fun_headline_variants":["Neural nets reveal helical paths for 3D active particles","Most probable path of active particle goes 3D helical","From U-shapes to helices: neural OMI solver finds new paths","3D active particle's optimal path becomes helix as time grows","Neural optimization uncovers helical transition paths"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The central assumption is that the trained neural networks reach the global minimum of the Onsager-Machlup integral for the helical paths; the paper provides no independent check that these paths are true global minima rather than local-minimum artifacts.","fun_headline_variants_meta":{"raw":{"variants":["Neural nets reveal helical paths for 3D active particles","Most probable path of active particle goes 3D helical","From U-shapes to helices: neural OMI solver finds new paths","3D active particle's optimal path becomes helix as time grows","Neural optimization uncovers helical transition paths"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000308,"raw_usage":{"total_tokens":1567,"prompt_tokens":683,"completion_tokens":884,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":427,"completion_tokens_details":{"reasoning_tokens":810}},"tokens_in":427,"tokens_out":884,"duration_ms":7504,"temperature":1.0,"reasoning_tokens":810,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T21:09:51.079553+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Euler-Lagrange residuals of the optimized helical paths; if the residuals do not vanish along the trajectory, or if retraining from many random initializations converges to different OMI values, then the reported H-path is not the true most probable path.","supporting_citations":[],"review_version":1}