On-the-Fly Path Planning for the Design of Compositional Gradients in High Dimensions

Ian McCue; Samuel Price; Zhaoxi Cao

REVIEW 3 major objections 5 minor 39 references

On-the-Fly Path Planning for the Design of Compositional Gradients in High Dimensions

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read This paper claims that replacing machine-learned surrogates with direct on-the-fly thermodynamic sampling—plus a relaxed bad-phase constraint and the Bad Phase Purge transformation—makes composition-gradient path planning tractable in…

desk verdict A genuinely useful engineering method, honestly presented, but the bad phase purge rests on an unproven continuity conjecture and the headline speedups lean on an estimated surrogate baseline. read the letter →

arxiv 2501.00012 v2 pith:6B3OVJBJ submitted 2024-12-13 physics.comp-ph cond-mat.mtrl-sci

Samuel Price , Zhaoxi Cao , Ian McCue This is my paper · ORCID

classification physics.comp-phcond-mat.mtrl-sci

keywords compositiongradientsfunctionallygradedmaterialsCALPHADrapidly-exploringrandomtreesBadPhasePurgeleverrulehigh-dimensionalpathplanningon-the-flysampling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that planning composition gradients in multi-element alloys does not require building a machine-learned surrogate of thermodynamic phase predictions. Instead, querying the thermodynamic calculation directly during a rapidly-exploring random tree search, while temporarily allowing bad phases, finds feasible paths with orders of magnitude fewer calculations. The relaxed path is then pushed out of bad-phase regions by the Bad Phase Purge formula, a lever-rule reweighting of equilibrium phases. In benchmarks on four structural alloys, the approach was up to about a million times faster than surrogate modeling, and it succeeded in cases where strict planning stalled.

What carries the argument

The load-bearing object is the Bad Phase Purge (BPP) transformation, defined as $$\$\sigma$^*(\$\alpha$) = H_2(\$\sigma$(\$\alpha$)) \cdot \frac{1}{H_1(\$\sigma$(\$\alpha$)) \cdot \vec{V}_{\mathrm{good}}} \left( H_1(\$\sigma$(\$\alpha$)) \odot \vec{V}_{\mathrm{good}} \right),$$ where $H_1$ maps a composition to its equilibrium phase fractions, $H_2$ maps a composition to a matrix of the equilibrium phase compositions, and $\vec{V}_{\mathrm{good}}$ is a 0/1 vector marking acceptable phases. The formula uses the lever-rule fact that, within a multiphase region, all compositions on a tie-simplex share the same phase compositions, so renormalizing the good-phase fractions moves the point out of bad-phase territory without altering the good phases' compositions. Applying this pointwise to a discretized path needs no additional thermodynamic queries. The paper proves each transformed point has zero bad phase by a short algebraic identity; it states but does not prove that the transformed path remains continuous.

What would settle it

Take one of the relaxed paths from the paper, apply Bad Phase Purge to adjacent discretized points, and evaluate the equilibrium phase fractions along the straight interpolation between each purged pair: if any interpolated segment shows a nonzero bad-phase fraction, or if adjacent purged points jump across a bad-phase region, then the purge does not guarantee a feasible gradient.

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Extended reading notes

Core claim

The central discovery is that the bottleneck in computational gradient design is the upfront surrogate, not the planner: in systems with more than four elements, the number of ground-truth thermodynamic (CALPHAD) calculations needed to build a surrogate grows exponentially with element count, whereas the number of direct queries an RRT needs to find a first feasible path depends on the visibility of the free space, not the dimension. The paper's method therefore samples the equilibrium calculation on the fly during RRT growth, relaxes the bad-phase constraint to widen narrow corridors, and then applies the Bad Phase Purge transformation pointwise to eliminate bad phases from the resulting path. For the six alloy pairs tested, this relaxed-plus-purge pipeline found paths with effectively zero bad phase everywhere, including two Al-rich pairs where the strict constraint found no path within two weeks. The reported speedup over surrogate modeling ranges from about 8×$10^{2}$ to $10^{6}$ across the test systems.

Load-bearing premise

The whole method rests on the premise that applying the Bad Phase Purge separately to each sampled point of a relaxed path leaves a connected, bad-phase-free path, but the paper proves only that each transformed point is free of bad phases, not that the transformed path is continuous.

Editorial extensions

If this is right

For the four structural alloys tested, every relaxed path after Bad Phase Purge had effectively zero bad phase at all sampled points, including alloy pairs where a strict RRT could not find a path within two weeks.
The number of thermodynamic calculations for the first feasible path scales with the geometry of the feasible region rather than with the number of elements, so the advantage over surrogate modeling grows as systems become more element-rich.
With RRT*, Bad Phase Purge can be applied on-the-fly, so optimized paths for path length or integrated chemical-potential gradient are evaluated on the purged, strict-constraint path rather than on the relaxed path.
Because Bad Phase Purge works pointwise and uses only data already returned by the equilibrium calculation, the total thermodynamic cost of the relaxed-plus-purge pipeline is essentially the number of RRT collision checks.

Reading between the lines

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

The paper leaves implicit that the same relax-and-purge strategy could apply to other equilibrium constraints: any quantity defined through phase fractions and phase compositions can be assigned a good/bad mask, so the recipe is not tied specifically to 'bad phases'.
A natural extension is to cache the $H_1$ and $H_2$ outputs from earlier collision checks and reuse them for nearby queries; because Bad Phase Purge is pointwise and costs no extra thermodynamic evaluations, this hybrid could preserve the speedup while cutting duplicate calculations.
The reported speedup depends on the surrogate training-cost estimate used for comparison; if future active-learning surrogates need far fewer than $5\times10^d$ samples, the dimension crossover at which direct sampling wins will shift.
If the unproved continuity of the purged path fails in some phase diagram, an adaptive-resolution repair is plausible: refine only segments whose purged endpoints lie in different phase regions, testing the interpolation at higher resolution before accepting the path.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the quick take: this is a real advance for practical composition-gradient design in 5–9 element systems, and the authors are refreshingly upfront about the one mathematical gap. The OTF approach—sampling CALPHAD directly during RRT instead of building a surrogate—is a natural but genuinely useful departure, and the relaxed-constraint + Bad Phase Purge trick is clever. Pointwise, the purge provably removes bad phases using lever-rule tie-simplex geometry, and the paper shows convincingly that it cuts CALPHAD queries by orders of magnitude on real alloy pairs.

Where it gets soft: the transformation is only conjectured to preserve path continuity, and the authors say so plainly in the Supplementary Notes. If adjacent purged points jump, the interpolated segment can re-enter bad-phase regions; Section 3.3 even footnotes small incursions. That is a genuine gap between “each sampled point is clean” and “the physical gradient is feasible.” For a manufacturing design tool, that matters. The speed-up numbers are also computed against an estimated surrogate-model cost (5×10^d) rather than a measured baseline for the high-dimensional cases, so the 10^6 figure is an upper-bound-ish estimate, not a direct measurement.

That said, the empirical evidence is decent: six alloy pairs, three seeds, paths verified to have <10^-2 bad phase at discrete points. The authors don't overclaim; they call the BPP a conjecture and invite counterexamples. The pointwise algebra is correct, there are no fitted parameters, and the citation pattern to the surrogate-model literature [10,11,17] is appropriate.

Who is this for? Someone building computational design tools for additive manufacturing or doing CALPHAD-based path planning. They'd get a new method worth trying. A serious referee should engage, mainly to push for either a proof of continuity under reasonable assumptions (e.g., within a single tie-simplex region or across boundaries with non-degenerate good-phase fractions) or a dense revalidation protocol that checks interpolated segments and quantifies incursions. I'd take it to peer review; it deserves referee time.

Referee Report

3 major / 5 minor

Summary. The paper proposes an on-the-fly path planning method for composition gradients in high-dimensional alloy systems, replacing upfront surrogate modeling of CALPHAD with direct equilibrium calculations during RRT sampling. To overcome narrow feasible corridors, the authors relax the bad-phase constraint and then apply a pointwise 'Bad Phase Purge' (BPP) transformation, based on the lever rule, to move each point along its equilibrium tie-simplex until no bad phases remain. The method is benchmarked on six pairs of structural alloys at 400 °C and on optimal planning with RRT*. The authors prove pointwise elimination of bad phases, but explicitly state they cannot prove continuity of the transformed path. Reported speed-ups over surrogate modeling range from 8x10^2 to 10^6.

Significance. If the continuity of the BPP-transformed path can be established or validated, the paper offers a practically valuable alternative to surrogate-based planning in 5-9 element systems: the transformation contains no fitted parameters, is derived from equilibrium thermodynamics, and the pointwise algebra in the Supplementary Notes is correct and machine-checkable. The paper is also honest in flagging the unproven part of the conjecture. However, the headline efficiency numbers are relative to an estimated, not measured, surrogate training cost, and the feasibility of the returned paths rests on the unproven continuity property, so the significance is conditional at this stage.

major comments (3)

[Appendix A / Supplementary Notes] The central feasibility claim depends on continuity of the transformed path, which the authors explicitly state they have not proven: 'We have been unable to prove (2).' Consequently, a straight-line segment between two adjacent purged points at the 0.01 discretization can re-enter bad-phase regions, and the footnote in §3.3 concedes that 'small incursions into bad or limited may occur between the discrete points.' Because the speed-ups in Figure 5 are reported for paths whose feasibility rests on this property, please either prove continuity (or a bound on incursion size) or perform a dense revalidation for all six paths and random seeds, reporting the maximum bad-phase fraction on interpolated sub-intervals between every pair of adjacent purged points.
[§3.3, Figure 5] The efficiency benchmark compares RRT+purge query counts against a surrogate-model training cost estimated as 5×10^d, cited to Ref. [17] as 'one example using at least 5×10^d samples.' This is an external estimate rather than a measured surrogate cost for the six alloy systems considered here; for d=9 it implies 5×10^9 CALPHAD calls, which this paper cannot verify. The 8×10^2 to 10^6 speed-up factors are therefore ratios to an unmeasured quantity. Please state explicitly that this is a literature-derived estimate, cite the exact location in Ref. [17], and, if possible, directly measure the surrogate construction cost for at least the 5-element Ti64-AlSi10Mg system.
[Eq. (1), Appendix A] Equation (1) is undefined whenever H1(sigma(alpha))·V_good = 0, i.e., when the equilibrium at a point contains no good phase. The conjecture assumes only f_undesirable < 1, which does not exclude such points because 'limited' phases count neither as good nor bad. In addition, the encoding of limited phases in V_good is not defined: Eq. (1) says 'good (1) or bad (0)', yet §3.3 and Figure 7 indicate that BPP removes limited phases as well. Please specify the encoding of limited phases, add the condition H1·V_good > 0 to the conjecture (or prove it for the generated paths), and report the minimum good-phase fraction along each of the six paths.

minor comments (5)

[§2] The steering distance is given as '√2/10, representing 10% of the maximal distance within any composition space'; please clarify the units and the definition of maximal distance, since the raw value is ambiguous in a simplex with normalized composition coordinates.
[§3.3] The sentence 'the speed-up provided by the new approach ranges from 8x10^2 (Ti64 to AlSiMg) to 10^6 (Ti64 to AlSiMg)' lists the same alloy pair twice; one of these should likely be a different pair, such as IN718-AlSiMg.
[Supplementary Notes] The line 'Similarly, we can show H1(sigma*)·Vgood =' is incomplete and lacks the right-hand side of the identity; please complete this displayed equation.
[Figure 8] The caption refers to 'ten unique pairs of five alloys (the four alloys described previously along with NiTi)', but NiTi is not defined in Table 1 with a composition or limited-phase list; please add this information or remove NiTi from the figure.
[General] The manuscript does not include a data or code availability statement; providing the Thermo-Calc scripts, the relaxation threshold used for each run, and the random seeds would greatly improve reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the BPP transformation is an original conjecture evaluated against CALPHAD ground truth, and the surrogate-model comparison uses an external literature estimate rather than a fitted parameter.

full rationale

The paper's central derivation chain is self-contained against an external ground truth (Thermo-Calc CALPHAD). The Bad Phase Purge transformation, Eq. (1), is not fitted to the target outcome: it is a pointwise map defined from the equilibrium phase-fraction and phase-composition functions H1 and H2 at each sampled composition, and the claim that the transformed point has zero bad phase is a stated conjecture (Appendix A) whose algebraic part is proved in the Supplementary Notes under the tie-simplex assumption and whose empirical part is checked by direct CALPHAD evaluation (Figure 4). The paper explicitly flags the unproven continuity of the transformed path and possible small incursions between discretized points (Section 3.3 footnote), which is an admitted correctness/completeness gap rather than a circular dependence. The speed-up comparison uses the surrogate-model training cost 5x10^d from prior literature [17] as an external baseline, not as a parameter fitted to the present results; the numerator is measured CALPHAD-query counts from the RRT runs. No load-bearing self-citations appear in the reference list, and the relaxed-RRT/BPP claim does not reduce to its own inputs by construction: the transformation could have failed to produce zero bad phases at the new compositions, and the paper reports that it did not. Accordingly, no circular step meeting the quoted-evidence standard was found.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The BPP transformation introduces no fitted parameters, but the efficiency benchmark depends on a hand-chosen 0.8 bad-phase allowance and on a literature-derived surrogate sampling estimate. The only genuinely unproved input is the continuity of the purged path, which is acknowledged as a conjecture.

free parameters (3)

Relaxed bad phase constraint = 0.8
The paper allows up to 80% total bad phase during RRT search to widen corridors; the exact value is chosen by hand and directly affects the number of CALPHAD queries.
RRT steering distance = √2/10
Set to 10% of the maximal distance in the composition simplex; affects how many collision checks are made.
Edge discretization resolution = 0.01
Edge collision checking step in composition space; controls BPP sampling density and whether incursions slip between points.

assumptions (4)

domain assumption Multiphase equilibrium compositions define a tie-simplex, and moving within the tie-simplex changes only phase fractions (lever rule behavior).
Used to justify Equation (1) for purging bad phases; standard thermodynamics, cited to [21].
ad hoc to paper The Bad Phase Purge transformed path is continuous and stays in good-phase regions between discretization points.
Stated as a conjecture in Appendix A; the authors say they have been unable to prove continuity and have not found a counterexample.
domain assumption CALPHAD equilibrium predictions using TCFE9 are a valid ground truth for feasibility classification.
The collision-checking function classifies compositions based on CALPHAD equilibrium phases; accuracy is not validated against experiments here.
domain assumption The number of ground truth samples needed to build a surrogate model scales as at least 5×10^d, per ref [17].
Used as the baseline to compute speed-ups; the authors do not build surrogates for most systems because it would be intractable.

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Pith. "Pith review of On-the-Fly Path Planning for the Design of Compositional Gradients in High Dimensions." pith.science (2026). https://pith.science/paper/6B3OVJBJ

@misc{pith2026250100012,
  author       = {Pith},
  title        = {Pith review of: On-the-Fly Path Planning for the Design of Compositional Gradients in High Dimensions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6B3OVJBJ}},
  note         = {Machine review of arXiv:2501.00012}
}

abstract

Functional gradients have recently experienced an explosion in activity due to advances in manufacturing, where compositions can now be spatially varied on-the-fly during fabrication. In addition, modern computational thermodynamics has reached sufficient maturity -- with respect to property databases and the availability of commercial software -- that gradients can be designed with specific sets of properties. Despite these successes, there are practical limitations on the calculation speeds of these thermodynamic tools that make it intractable to model every element in an alloy. As a result, most path planning is carried out via surrogate models on simplified systems (e.g., approximating Inconel 718 as Ni$_{59}$Cr$_{23}$Fe$_{18}$ instead of Ni$_{53}$Cr$_{23}$Fe$_{18}$Nb$_{3}$Mo$_{2}$Ti$_{1}$). In this work, we demonstrate that this limitation can be overcome using a combination of on-the-fly sampling and a conjectured corollary of the lever rule for transformations of isothermal paths in arbitrary compositional dimensions. We quantitatively benchmark the effectiveness of this new method and find that it can be as much as 106 times more efficient than surrogate modeling.

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Reference graph

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Let 𝑍𝑍 �Δ𝑑𝑑−1, standard simplex, where 𝑑𝑑 is the number of elements in the system� denote the composition space
[35]

Let 𝑧𝑧𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 ∈ 𝑍𝑍 be the starting composition and 𝑧𝑧𝑒𝑒𝑒𝑒𝑑𝑑 ∈ 𝑍𝑍 be the ending composition
[36]

Let a composition gradient be represented by 𝜎𝜎: [0,1] → 𝑍𝑍 a continuous function such that 𝜎𝜎(0) = 𝑧𝑧𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 and 𝜎𝜎(1) = 𝑧𝑧𝑒𝑒𝑒𝑒𝑑𝑑
[37]

Let 𝑓𝑓𝑢𝑢𝑒𝑒𝑑𝑑𝑒𝑒𝑠𝑠𝑢𝑢𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢𝑒𝑒: 𝑍𝑍 → [0,1] be a function that maps a composition to the total amount of undesirable phases present in equilibrium (at the isothermal temperature) We conjecture: 𝑓𝑓𝑢𝑢𝑒𝑒𝑑𝑑𝑒𝑒𝑠𝑠𝑢𝑢𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢𝑒𝑒(𝑧𝑧𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠) = 0, 𝑓𝑓𝑢𝑢𝑒𝑒𝑑𝑑𝑒𝑒𝑠𝑠𝑢𝑢𝑠𝑠𝑠𝑠𝑢𝑢𝑢𝑢𝑒𝑒(𝑧𝑧𝑒𝑒𝑒𝑒𝑑𝑑) = 0, an...
[38]

All transformed points are entirely not bad (i.e., completely good): 𝐻𝐻1�𝜎𝜎∗(𝛼𝛼)� ∙ V��⃗bad = 0
[39]

We have been unable to prove (2)

The transformed path, 𝜎𝜎∗(𝛼𝛼), is continuous: We can prove (1): We have that 𝐻𝐻1�𝜎𝜎∗(𝛼𝛼)� = 1 �𝐻𝐻1�𝜎𝜎(𝛼𝛼)� ∙ V��⃗good� �𝐻𝐻1�𝜎𝜎(𝛼𝛼)� ⊙ V��⃗good� Thus, 𝐻𝐻1�𝜎𝜎∗(𝛼𝛼)� ∙ V��⃗bad = 1 �𝐻𝐻1�𝜎𝜎(𝛼𝛼)� ∙ V��⃗good� �𝐻𝐻1�𝜎𝜎(𝛼𝛼)� ⊙ V��⃗good� ∙ V��⃗bad By the properties of the Hadamard produc...

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

[1] [17]

Silvestroni, D

L. Silvestroni, D. Pavan, C. Melandri, D. Sciti, N. Gilli, L. Ortiz- Membrado, E. Jiménez-Piqué, A.M. Grande, Functionally graded ultra-high temperature ceramics: From thermo-elastic numerical analysis to damage tolerant composites, Mater. Des. 224 (2022) 111379. https://doi.org/10.1016/j.matdes.2022.111379

arXiv 2022

[2] [1]

good” (green), “limited

The amount of total bad phase is effectively zero (<10-2) at every point, while the limited phases only appear at the ends of the gradients † and in fractions less than or equal to their amount in the base alloys. Before the RRT starts in earnest, the linear composition profile between the two compositions is tested . If it is feasible, then the run stops...

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[3] [2]

Surrogate models scale poorly when designing functional gradients that account for every element in the alloy systems of interest

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[4] [3]

On-the-fly sampling can reduce the number of CALPHAD queries by orders of magnitude compared to surrogate models

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[5] [4]

Here, the number of collision checks was reduced by relaxing the constraints (i.e., allowable fraction of bad phases)

The efficiency of on-the-fly sampling can be further enhanced by reducing the number of collision checks that the path planning algorithm must make . Here, the number of collision checks was reduced by relaxing the constraints (i.e., allowable fraction of bad phases)

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[6] [5]

A conjecture is provided to demonstrate this concept is generalizable, and we have yet to find an example where it does not hold

While this strategy appears to be at odds with gradient design goals, it can be remedied by making use of phase equilibria properties (i.e., the Lever Rule). A conjecture is provided to demonstrate this concept is generalizable, and we have yet to find an example where it does not hold

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[7] [6]

break -even

We have also demonstrated that this approach can be extended to the challenge of designing gradients that optimize a particular cost function such as path length and integrated ∇µ by using the RRT* algorithm and performing the BPP operation on-the -fly. In fact, th is can be done with any cost function that can be calculated from CALPHAD-derive d properti...

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[8] [7]

The algorithmic foundations of these hybrid methods for collision checking have already been developed in the path planning community [23]

A hybrid sampler , which stores the results generated by on-the -fly sampling and uses them to predict the results of future queries if they lie sufficiently close to the previous known results, could provide the best of both on -the-fly sampling and traditional surrogate modeling. The algorithmic foundations of these hybrid methods for collision checking...

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