{"id":"27bfe078-cc2a-4d56-bc93-cc7ac404fed8","arxiv_id":"2506.19550","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"A symbolic-regression loss based on the Lie symmetry condition finds exact symmetry generators for ten nonlinear ODE systems where the SYM package fails.","lead":"This paper uses symbolic regression to automatically discover Lie point symmetries of systems of ordinary differential equations. This could help solve nonlinear ODEs that computer algebra systems often struggle with.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'existing CASs cannot find' claim rests on a single package (SYM) in a single configuration; if another CAS or a different SYM ansatz solves any of the ten problems, the headline superiority claim is falsified.","rationale":"The reader's weakest_assumption already identified the CAS-representativeness issue as an additional assumption behind the broad claim. My stress-test agrees and makes it the primary load-bearing concern: the paper's headline superiority claim is a comparative statement about 'existing CASs,' yet the evidence is one package in one configuration. The symbolic verification of the ten found generators is real and valuable evidence that the method works on those examples, so I do not call the method unsound or reject the paper. The appropriate verdict remains CONDITIONAL: the paper should temper the 'existing CASs cannot find' wording or broaden the benchmark to at least one independent CAS. The finite-sample loss issue is secondary; it would matter for claims of general reliability, but the paper's concrete ten-for-ten demonstrations are verified post hoc, so they stand. The proposed concrete test directly probes the comparative claim: if another CAS solver succeeds on any of the ten, the central claim needs revision; if not, the claim is better supported. No change to the reader's CONDITIONAL verdict is needed.","tokens_in":16121,"tokens_out":14167,"duration_ms":160167,"concrete_test":"Run the determining-equation systems for ODE8-10, the cases where even SYM-S fails, through an independent CAS symmetry solver such as Maple's PDEtools:-DeterminingPDE followed by pdsolve, and through Mathematica's DSolve with its built-in symmetry method or SYM with alternative ansatz forms (e.g., bounded-degree polynomial or rational ansatz). If any solver returns a valid symmetry generator for any of the ten systems, the statement 'existing CASs cannot find' is false as stated; if none do, the comparison is materially strengthened.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The abstract claims symbolic regression finds symmetries 'that existing CASs cannot find,' and the Discussion narrows this to a comparison against Mathematica's SYM package in three fixed modes. The ten examples are credible: the found generators are listed in Table 4 and are verified symbolically, which rules out false positives in the reported results. However, the load-bearing external claim is broader than the evidence. Only one CAS package is tested, and the paper gives no analysis of why SYM's determining-equation PDEs are unsolvable: the failures could stem from the specific ansatz choices, the version of Mathematica/SYM, or the particular solving strategy rather than an intrinsic limitation of computer algebra symmetry finding. The paper itself notes that MathLie requires user interaction and Maple's symgen cannot handle systems, but it does not test Maple's PDEtools or alternative SYM configurations. Additionally, the finite-sample loss L_D,f is evaluated only on points from three RK trajectories; a zero there does not imply the continuous symmetry condition, so the automated search can, in principle, return a spurious zero-loss expression on a new ODE. Symbolic verification catches such outputs only after the search, so the method's general reliability is not certified. The central claim that the method 'enlarges the space of ODEs' whose solutions can be approached depends on the failures being intrinsic, which the current single-package comparison does not establish.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"Kahlmeyer et al. propose a search-based symbolic regression method for discovering Lie point symmetry generators of first-order ODE systems. Given a symbolic ODE y' = f(t,y), they simulate RK45 trajectories, then minimize a loss L_{D,f} that enforces the reduced symmetry condition (Eq. 1) at the sampled points, using the DAG-based multi-output symbolic regressor from Kahlmeyer et al. (2024). The Hamiltonian/time-evolution component of the generator is removed so that the search returns only informative, non-redundant symmetries. The paper demonstrates the method on ten nonlinear 2D ODE systems, reporting for each a closed-form generator with near-zero numerical loss and successful symbolic verification (Table 4). It compares with Mathematica's SYM package in automatic and two interactive modes (Table 1), on which SYM mostly fails, and it illustrates the use of two independent generators on one example system.","tokens_in":16390,"tokens_out":10679,"duration_ms":113556,"significance":"Within a properly scoped claim, this is a solid contribution. The algebraic derivation of the reduced symmetry condition (Eq. 1) is correct, and the removal of the trivial Hamiltonian part is algebraically justified. The strongest evidence is Table 4: every reported generator is validated symbolically, not merely by finite-sample loss, so the ten reported findings are not false positives. The circularity concern raised in the stress test does not land, because the known generators enter only through verification and not through the optimized loss. The idea of using a modern multi-output symbolic regressor directly on the Lie condition is a natural but useful adaptation, and the code is released. If the external comparison to computer algebra systems is qualified to the specific package and modes tested, the paper is a meaningful step toward automatic symmetry discovery for nonlinear ODEs.","major_comments":[{"comment":"The headline claim that the method finds symmetries 'that existing CASs cannot find' is broader than the evidence presented. Only one package (Mathematica's SYM) in three fixed modes is tested, and the paper gives no details about the SYM version, solving time, or the precise ansatz used in the interactive modes. Table 1 shows that SYM-S, which supplies the known generator as a starting ansatz, already succeeds on seven of the ten problems, so the failures are configuration-dependent rather than an intrinsic property of computer algebra symmetry finding. I recommend either (a) testing at least one further independent symmetry-finding implementation, e.g., Maple's PDEtools-based facilities, or (b) rewriting the title, abstract, and conclusion to say 'symmetries that are not found by Mathematica's SYM package in the configurations described here'.","section":"Abstract; Discussion, 'Hard Problems for Mathematica and SYM'"},{"comment":"The statement that the Hamiltonian symmetry 'cannot be used for simplifying the ODE system' is not correct as stated. For the introductory system f = (-y2, y1), canonical coordinates for X_H = ∂t - y2∂y1 + y1∂y2 give r = sqrt(y1^2 + y2^2), v = t, and s = atan(y1/y2), and the reduced system is r' = 0, s' = 1, which is a genuine order reduction. The exclusion of the Hamiltonian component is better motivated by the search objective of avoiding rediscovery of f, not by an inability to simplify. Please correct or clarify this claim, because it is used to justify restricting the search to generators of the form <η*, ∇y>.","section":"Section 'Hamiltonian Symmetry'"},{"comment":"The method's general reliability is not certified by the finite-sample loss. L_{D,f} is zero if Eq. (1) holds on finitely many points of a few RK trajectories, and a zero there does not imply the continuous PDE condition. In the reported experiments the subsequent symbolic verification in Table 4 is what rules out false positives, and this should be stated explicitly as part of the method pipeline. The conclusion's sentence that the approach 'enlarges the space of ODEs' whose analytical solution can be approached extrapolates beyond the evidence; a limitations paragraph is needed to describe the sample dependence and the heuristic nature of the search.","section":"Section 'Symbolic Regression for Symmetry Finding'; Discussion and Conclusion"}],"minor_comments":[{"comment":"The text 'see subsection' is a dangling reference; supply the subsection number or title.","section":"Supplementary Material, 'Identifiability'"},{"comment":"The main text describes the data as a single sample D = {t_i, y(t_i)}, while the experiments use three time series (Table 3). State explicitly that D is the union of points over the multiple trajectories.","section":"Section 'Symbolic Regression for Symmetry Finding'"},{"comment":"The column 'Symbolic Loss' with entries 'True/True' should explain the verification procedure, e.g., simplification of the expression to zero by SymPy, and the threshold or simplification method used.","section":"Table 4"},{"comment":"Figure 5 is hard to read because running times and expression sizes are combined in one panel with two different scales; separate panels or explicit dual-axis labeling would improve clarity.","section":"Figure 5"},{"comment":"The reference to 'Richardson and Dale. 2014' is incomplete; include the full author list and title of the book chapter.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"To the editor: the mathematical core and the ten verified results are credible. My main concern is the breadth of the 'existing CASs cannot find' claim and the unqualified generalization from a single SYM configuration. Revision can fix this either by adding at least one more CAS comparison or by carefully scoping the title, abstract, and conclusion. I would not accept the paper in its current form."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, the core demonstration is real: on ten nonlinear 2D ODE systems where Mathematica's SYM package fails, the DAG-based symbolic regressor finds Lie point symmetry generators, and every one is verified symbolically (Table 4). Second, the headline claim that existing CASs \"cannot find\" these symmetries is broader than the evidence: only SYM in three fixed modes was actually run.\n\nThe genuinely new piece is the loss function: instead of learning conserved quantities or coordinate transformations with a neural network as in AI Poincaré or Liu & Tegmark, the authors directly minimize the Lie symmetry condition residual over expression DAGs, with the Hamiltonian symmetry removed to avoid the trivial time-evolution generator. The derivation of the reduced condition (Eq. 1) is correct, and removing the Hamiltonian part is a sensible way to cut down the search space and make independent generators meaningful. The independent-generator demo, where two generators lead to two different reductions that give the same solution, is a nice showing.\n\nCredit where due: the symbolic verification rules out false positives, and the authors use three time series per system, matching their own identifiability argument for n=2. Code is on GitHub. The paper is honestly written about its scope in the Discussion.\n\nSoft spots, in proportion. The CAS comparison is narrow. The paper justifies it by saying Maple's symgen can't handle systems and MathLie needs user interaction, but that is assertion, not a test. A different CAS, or a different SYM ansatz, might solve some of the ten. That doesn't undercut the positive result—the method clearly finds valid generators—but it does undercut the \"enlarges the space\" claim as stated. I'd ask them to either temper the abstract or run the same problems through Maple's PDEtools and another SYM configuration. Minor: the finite-sample loss means a zero on sampled points doesn't guarantee the continuous condition; symbolic verification catches this after the fact, so for a new ODE the method is a heuristic, not certified. Also the DAG search hyperparameters (operator set, size budget) are not specified; the code is available, but for reproducibility they should report them.\n\nThe identifiability caveat in the supplement is honest, and the chosen data set satisfies it, so I don't count that as a flaw.\n\nWho this is for: people building symbolic regression tools for differential equations, and the symmetry-method community wanting a complement to CAS packages. It is an engineering advance with a solid demonstration, not a deep theoretical result.\n\nRecommendation: send to peer review. The CAS-comparison overstatement is fixable, the verification is credible, and the method deserves referee time. I would cite it if I worked on symbolic regression or automated symmetry finding. Reading group: maybe.","headline":"A solid, honestly-scoped heuristic for finding Lie point symmetries by symbolic regression; the method demonstrably works on ten nonlinear ODEs, but the 'existing CASs cannot find' claim is one package wider than the evidence.","tokens_in":16916,"tokens_out":4007,"would_cite":true,"duration_ms":38250,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34C14"],"pacs":[],"model":"deepseek-v4-flash","headline":"Search-based symbolic regression finds ODE symmetries that computer algebra systems miss.","keywords":["Lie point symmetries","ordinary differential equations","symbolic regression","Hamiltonian symmetry","order reduction","nonlinear ODEs","determining equations","expression DAG"],"falsifier":"Run the pipeline on a nonlinear ODE whose known symmetry generator is too complex for the search budget: if the sampled loss reaches machine precision on the training curves but the best expression fails the exact symmetry condition at densely sampled off-trajectory points, then a small sampled loss is not sufficient evidence of a true global symmetry. A complementary check is to feed the found generator back into an exact symbolic verifier on a held-out set of initial conditions and see whether the reduced ODE system still solves the original problem.","tokens_in":15905,"feed_emoji":"🧮","tokens_out":7801,"duration_ms":80172,"temperature":0.7,"pith_summary":"This paper tries to establish that search-based symbolic regression can replace the PDE-solving step in Lie symmetry analysis of ODEs. The authors build a loss function from the continuous symmetry condition, evaluate it on a few numerically simulated solution curves, and search over symbolic expression graphs for the components of a symmetry generator. They report that this recovers simple symmetry generators for ten nonlinear systems for which an established computer algebra symmetry finder does not succeed, with every recovered generator verified both numerically and symbolically. If this holds, symmetry-based order reduction becomes available for a wider class of nonlinear ODE systems, because the bottleneck moves from solving determining partial differential equations to searching expression space.","feed_headline":"Symbolic regression finds ODE symmetries that CASs miss","feed_subtitle":"A Lie-symmetry loss guides expression search to generators on ten nonlinear systems that stalled a computer algebra symmetry finder.","key_machinery":"The load-bearing object is the simplified symmetry condition: after removing the Hamiltonian part, a generator $\\eta^\\star$ is valid exactly when $$0 = \\nabla \\eta^\\star(t,y)\\cdot \\binom{1}{f(t,y)} - \\nabla f(t,y)\\cdot \\binom{0}{\\eta^\\star(t,y)},$$ with gradients taken over $t$ and $y$. The paper's loss $L_{D,f}(\\eta^\\star)$ is the average squared residual of this identity over sample points $D$ from Runge-Kutta trajectories; it is zero when the sampled points satisfy the condition. The search is carried out by a multi-output symbolic regressor that enumerates directed acyclic graph (DAG) skeletons up to a size budget, labels operator nodes with function symbols, computes gradients by automatic differentiation, and discards near-zero generators to avoid the trivial solution $\\eta^\\star=0$.","core_discovery":"The central claim is that the difficult part of symmetry finding, solving the overdetermined PDE system for the generator components, can be bypassed by searching over candidate generator functions with a loss that directly encodes the Lie condition. For a first-order system $y'=f(t,y)$, the paper removes the always-present Hamiltonian symmetry $X_H=\\partial_t+\\langle f,\\nabla_y\\rangle$, so only the informative part $\\eta^\\star$ remains; the corresponding reduced symmetry condition is a pointwise algebraic equation involving $\\eta^\\star$, its gradients, and $f$. A candidate generator is scored by summing squared residuals of this condition over points from Runge-Kutta trajectories, and a multi-output DAG-based symbolic regressor searches for minimizers. The paper reports that this pipeline finds simple generators for all ten tested systems, and that each found generator satisfies the condition both numerically and symbolically.","pith_inferences":["The paper leaves implicit that the loss could be sampled on random points in state space instead of trajectory points; if that works, the method would depend only on the vector field $f$ and not on choosing informative initial conditions.","Because the search space of expression DAGs grows exponentially with generator complexity, the practical advantage over algebraic methods may be limited to symmetries with short formulas; one could test this by adding a complexity penalty to the loss.","The same loss is continuous in $\\eta^\\star$, so the pipeline could in principle detect approximate or partial symmetries on systems that have no exact Lie point symmetry, giving a quantitative measure of symmetry breaking.","The Hamiltonian-subtraction trick may transfer to symmetry finding for partial differential equations, where the analogous time-evolution symmetry is equally uninformative."],"forward_implications":["Symmetry finding no longer depends on a computer algebra system solving the determining PDEs; any system whose symmetries have compact symbolic generators becomes reachable by search.","Once a generator is found, it provides a coordinate transformation that reduces the ODE dimension by one, as demonstrated on a nonlinear example where two independent generators lead to solvable reduced systems.","Explicitly removing the Hamiltonian symmetry makes meaningful independent generators discoverable, so several symmetries can be collected and used separately to simplify a system.","The approach extends to higher-order ODEs through the standard first-order reformulation, widening the class of equations whose symmetries can be found automatically."],"supporting_citations":[{"why":"Supplies the multi-output DAG symbolic regressor that the symmetry search is built on.","marker":"Kahlmeyer et al. 2024"},{"why":"Describes the symmetry-finding package that serves as the computer algebra baseline for the comparison.","marker":"Dimas and Tsoubelis 2005"},{"why":"Provides the Runge-Kutta integrator used to generate the trajectories on which the symmetry loss is evaluated.","marker":"Dormand and Prince 1980"},{"why":"Supplies automatic differentiation for computing gradients of candidate generators inside the loss.","marker":"Paszke et al. 2017"},{"why":"Background reference for the Lie point symmetry condition and the order-reduction procedure the method builds on.","marker":"Bluman and Kumei 1989"},{"why":"Used for the symbolic verification of the found generators reported in the results table.","marker":"Meurer et al. 2017"}],"fun_headline_variants":["Symbolic regression finds ODE symmetries CASs can't","No PDE solving: symbolic regression finds ODE symmetries","Lie-symmetry regression outdoes computer algebra","Data-driven search for ODE symmetries, no CAS"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole approach rests on the assumption that fitting the symmetry condition on a few numerically simulated solution curves is enough to make the search settle on a generator that satisfies the exact symmetry condition everywhere, not just on those curves.","fun_headline_variants_meta":{"raw":{"variants":["Symbolic regression finds ODE symmetries CASs can't","No PDE solving: symbolic regression finds ODE symmetries","Lie-symmetry regression outdoes computer algebra","Data-driven search for ODE symmetries, no CAS"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000667,"raw_usage":{"total_tokens":2985,"prompt_tokens":831,"completion_tokens":2154,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":447,"completion_tokens_details":{"reasoning_tokens":2088}},"tokens_in":447,"tokens_out":2154,"duration_ms":17767,"temperature":1.0,"reasoning_tokens":2088,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:31:58.114544+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the pipeline on a nonlinear ODE whose known symmetry generator is too complex for the search budget: if the sampled loss reaches machine precision on the training curves but the best expression fails the exact symmetry condition at densely sampled off-trajectory points, then a small sampled loss is not sufficient evidence of a true global symmetry. A complementary check is to feed the found generator back into an exact symbolic verifier on a held-out set of initial conditions and see whether the reduced ODE system still solves the original problem.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Describes the symmetry-finding package that serves as the computer algebra baseline for the comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Runge-Kutta integrator used to generate the trajectories on which the symmetry loss is evaluated."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies automatic differentiation for computing gradients of candidate generators inside the loss."},{"cited_title":"W.; and Kumei, S","cited_arxiv_id":null,"evidence_quote":"Background reference for the Lie point symmetry condition and the order-reduction procedure the method builds on."},{"cited_title":"P.; Paprocki, M.; Cert \\' k, O.; Kirpichev, S","cited_arxiv_id":null,"evidence_quote":"Used for the symbolic verification of the found generators reported in the results table."}],"review_version":2}