{"id":"323588f5-03ae-4a08-bf39-a116e884ce1a","arxiv_id":"2412.21049","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"FEX, a symbolic regression method, is applied to epidemiological data and produces accurate one-step-ahead fits, but the paper does not show that the discovered expressions are interpretable or that its forecasts beat strong baselines.","lead":"This paper applies a reinforcement-learning-based symbolic regression method called FEX to learn equations of disease spread from synthetic and real COVID-19 data. It claims high-accuracy fits and interpretable expressions, but the real-data validation is mostly visual and the discovered equations are not shown to recover known models.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Real-data 15-day forecast claim is unsupported: Eq. (3.5) assumes noise-free Markovian dynamics in (R,D,Q), yet no noise model, error bars, or baseline comparison is provided; a controlled synthetic test with hidden compartments and reporting noise is needed.","rationale":"The reader's verdict is CONDITIONAL, and my analysis agrees. The synthetic experiments (Section 4.1) are internally consistent: because the data is generated with the same Euler integrator and the FEX expression class can represent the exact model equations, the reported MSE of 1e-8 is plausible. The real-data experiment (Section 4.2), however, carries the paper's practical claim, and it is the least secure. The training loss (3.5) is a one-step residual that assumes an exact deterministic discrete map. For real COVID-19 data, this map is corrupted by reporting noise, delays, and unobserved state variables. With 85 training points and a high-capacity symbolic search, overfitting is a real risk. The paper does not report forecast error metrics, error bars, random seed sensitivity, or a simple baseline, and it omits preprocessing details (e.g., normalization) that are essential for interpreting the learned equations. These omissions make the 'closely aligns' claim unverifiable. The proposed test—a controlled synthetic experiment with hidden compartments and noise—would directly probe whether the method's real-data pipeline can generalize under the exact conditions the assumption in (3.5) ignores. If FEX fails this test, the real-data contribution collapses; if it passes, the concern is mitigated and the remaining issues are merely reporting quality.","tokens_in":8746,"tokens_out":10667,"duration_ms":107225,"concrete_test":"Simulate a SEIRD model where only R, D, and Q (active cases) are observed, as in Section 4.2. Add realistic reporting noise (e.g., Poisson noise on daily counts) to generate noisy observed series, and normalize the observed series in the same (unspecified) way as the paper. Train FEX on the first 85 days using the exact loss (3.5) with Euler step Delta = 1 day, then recursively forecast days 86-100. Compare forecast error (RMSE or MAPE) against a persistence baseline that repeats the last observed value, across 10 independent noise realizations. If FEX's forecast error is not consistently lower than the persistence baseline, or if learned expressions vary drastically across noise realizations, the real-data claim in Section 4.2 is not supported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim has two parts: (i) on synthetic SIR/SEIR/SEIRD data, FEX achieves test MSE about 1e-8, far below NN/RNN; (ii) on real Hubei COVID-19 data, FEX closely aligns with the 15-day prediction horizon. Part (i) is credible because the data is generated by Euler with the same step used in the training loss (3.5) and the expression class contains the exact right-hand sides; the main issue here is reproducibility, not correctness. Part (ii) is the load-bearing weakness. The loss (3.5) fits the one-step Euler residual under the assumption that the observed daily sequence satisfies x_{s+1} = x_s + f(x_s) exactly, with no reporting noise, no delays, and with (R,D,Q) a Markovian sufficient state. Real COVID-19 reporting violates all three: counts are noisy, case definitions changed, and the population includes unobserved exposed/infectious compartments. With only 85 training points, FEX's search over a large symbolic space can overfit the training residual; the paper gives no error bars, no sensitivity analysis, no alternative time-window validation, and no comparison to a trivial baseline. The learned equations in Section 4.2 are products of polynomials and sinusoids with no mechanistic interpretation, and the preprocessing (normalization) is not described, making the forecast plots unverifiable. If the 15-day forecast is not robust to these issues, the paper's conclusion that FEX is a powerful epidemiological tool is overstated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies the Finite Expression Method (FEX), a reinforcement-learning-based symbolic regression framework, to learn governing differential equations for epidemiological dynamics. On synthetic SIR, SEIR, and SEIRD data generated by Euler's method, FEX reports test MSE values on the order of 1e-8 to 1e-7, outperforming neural-network and RNN baselines. On real COVID-19 data from Hubei (first 85 days training, 15 days test), FEX yields explicit expressions for dR/dt, dD/dt, dQ/dt and is compared visually with the fractional-order SEIQRDP model. The paper argues that FEX combines interpretability with high predictive accuracy and is a versatile tool for infectious-disease modeling.","tokens_in":9145,"tokens_out":2488,"duration_ms":25273,"significance":"Symbolic regression for epidemiological dynamics is a timely and potentially valuable direction, and the FEX framework itself is nontrivial: it uses RL-based combinatorial search and provides explicit expressions rather than black-box predictors. The synthetic experiments are clean and reproducible in principle, and the reported accuracy on the exact Euler-generated trajectories is strong evidence that FEX can recover known polynomial right-hand sides. However, the paper's central claim has two parts, and only the synthetic part is convincingly supported; the real-data forecast claim is not substantiated with quantitative metrics, error bars, or robustness checks. If the real-data results were properly validated, the contribution would be of interest to the machine-learning-for-dynamical-systems community. As it stands, the paper is a useful demonstration of FEX on controlled compartmental models, but its conclusion about practical epidemiological utility is overstated.","major_comments":[{"comment":"The real-data evaluation is not statistically supported. The loss (3.5) assumes the observed daily sequence satisfies x_{s+1} = x_s + f(x_s)Δ exactly, with no reporting noise, no delays, and with (R,D,Q) as a Markovian sufficient state; real COVID-19 reporting violates all three assumptions. The paper reports only visual alignment for the 15-day forecast and provides no error bars, no sensitivity analysis, no alternative time-window validation, and no comparison to a trivial baseline (e.g., persistence or a simple linear autoregressive model). The claim in Section 4.2 that FEX 'closely aligns with observed data' and 'demonstrates robust generalization' is therefore load-bearing but unsupported.","section":"§4.2, Eq. (3.5)"},{"comment":"The synthetic comparison is favorable to FEX by construction. The data is generated with Euler's method at the same time step Δ used in the training loss (3.5), and the operator library {0,1,x,x^2,x^3,x^4,sin,cos,exp,+,-,×} contains the polynomial terms (e.g., S·I, E, I) appearing in the SIR, SEIR, and SEIRD right-hand sides. Consequently, the reported MSE of order 1e-8 demonstrates that the optimizer can recover an expression class that is exactly present in the search space, but it does not establish that FEX outperforms NN/RNN under realistic noisy or misspecified dynamics. Without experiments that add observation noise, use a different integrator for data generation, or test with an incomplete operator library, the claimed 'significant outperforming' (Section 4.1, Numerical Results) is overstated.","section":"§4.1, Data Generation and Training Procedures"},{"comment":"The learned equations for dR/dt, dD/dt, and dQ/dt are products of polynomials and sinusoids with many coefficients, and the paper offers no mechanistic interpretation or parsimony analysis. The claim in the abstract and introduction that FEX 'uncover[s] explicit relationships among epidemiological variables' and provides 'interpretability' is not substantiated, because the recovered expressions are not related to any known epidemiological structure and no analysis of their biological plausibility is given. Additionally, the paper does not describe the normalization or preprocessing applied to the real data, making the exact equations and the forecast plots unverifiable.","section":"§4.2, Numerical Results"}],"minor_comments":[{"comment":"The action space of the controller is described only informally; for reproducibility, the paper should define precisely how positions in the tree map to the distributions p1_Φ, ..., ps_Φ and how the tree depth and arity constraints are enforced.","section":"§2, Operator Sequence Generation"},{"comment":"The hyperparameters T1, T2, T3, K, and the batch size for the RL controller are never given numerical values; without these, the experiments cannot be reproduced.","section":"§2, Candidate Optimization"},{"comment":"The RNN architecture is described as two LSTM layers with hidden size 51, but no details are given on sequence length, backpropagation through time, or whether the loss is computed on one-step or multi-step predictions, which is needed to interpret the comparison.","section":"§4.1, RNN description"},{"comment":"The dataset is said to span 1,147 days, but only the first 100 days are used; please clarify why this specific window is chosen and give the exact date range used for training and testing.","section":"§4.2, Data Acquisition"},{"comment":"Reference [31] (Teobaldi et al., PNAS) does not appear to be cited in the body of the paper; please remove it or cite it appropriately.","section":"References"},{"comment":"The notation in Eq. (3.5) uses the index i both for the component of the state and for the component-specific surrogate; consider using a different index (e.g., k) to avoid confusion.","section":"Eq. (3.5)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a competent demonstration of FEX on synthetic compartmental models, but the real-data section is the weak point. I would encourage the editor to request a proper real-data validation: quantitative forecast errors, comparison with a simple baseline, and robustness to data noise and preprocessing choices. If the authors cannot provide such validation, the paper's claims should be substantially narrowed to the synthetic setting. Also note that the paper is heavily self-referential (citing several FEX papers by the same group), which is not problematic per se, but the novelty relative to those prior works should be made explicit."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is an application paper, not a new method. The FEX framework is prior work by the same group; what is new is the epidemiological testbed: SIR/SEIR/SEIRD synthetic data and Hubei COVID-19 data. On the synthetic side the paper is basically sound. The component-wise Euler residual loss is clearly stated, the data-generation protocol is described, and the achieved test MSE of 1e-8 to 1e-7, well below NN and RNN, is consistent with what FEX should do when the target functions are in the expression class and the data is generated with the same integrator. That is a feature of the benchmark design, not a mystery. The paper earns credit for reporting the learned equations explicitly.\n\nSoft spots are on the real-data side. The 15-day Hubei forecast is supported only by a visual comparison with SEIQRDP: no error bars, no sensitivity to the training window, no quantitative metric, no code release. The loss assumes the daily counts satisfy x_{s+1}=x_s+f(x_s)Delta exactly, with no reporting noise, no delays, and with (R,D,Q) as a Markovian sufficient state; real COVID-19 data violates all three. With 85 training points and a large symbolic search space, the learned expressions can easily overfit the residual; the displayed equations are products of polynomials and sinusoids that do not have a mechanistic reading, which undercuts the interpretability claim. The paper's own limitations section mentions computational cost and nonuniqueness but not the noise/state-sufficiency issue. That is an omission, not a fatal flaw: the claims are stated as 'closely aligns' and 'highlights FEX as a powerful tool,' which is more than the evidence supports.\n\nWho this is for: readers working on symbolic regression for dynamical systems, or epidemiologists who want to see what off-the-shelf FEX can do. It deserves peer review because the method is real and the synthetic experiments are reproducible in principle, but the revision should add uncertainty quantification, a trivial baseline (e.g., constant or linear extrapolation) on the real data, a sensitivity analysis over the training window, and code. Without those, the real-data conclusion should be downgraded.","headline":"FEX is a real method with credible synthetic results, but the real-data forecasting claim is unsupported as presented.","tokens_in":9615,"tokens_out":1912,"would_cite":false,"duration_ms":19613,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["92D30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that the Finite Expression Method, a reinforcement-learning-based symbolic regression approach, learns explicit differential equations of epidemiological dynamics from data, achieving test errors around…","keywords":["finite expression method","symbolic regression","reinforcement learning","epidemiological dynamics","SIR SEIR SEIRD","COVID-19 forecasting","interpretable machine learning"],"falsifier":"Take the synthetic SIR/SEIR/SEIRD generators used in the paper, add Gaussian observation noise with standard deviation $10^{-3}$ to the training trajectories, and run FEX with the same settings; if the recovered expressions no longer approximate the true right-hand sides or test MSE rises above $10^{-4}$, the central accuracy claim holds only for noiseless data. For the real-data claim, retrain on the first 70 days of the Hubei series and forecast days 71--100; if the 30-day forecast diverges from reported counts much faster than the 15-day forecast, the learned equations are fitting short-term trends rather than the true dynamics.","tokens_in":8554,"feed_emoji":"🦠","tokens_out":6029,"duration_ms":55489,"temperature":0.7,"pith_summary":"Epidemic models usually come from expert-written compartmental equations, while neural-network predictors are accurate but opaque. This paper argues that the Finite Expression Method (FEX) gets both: it searches over symbolic expressions using reinforcement learning and fits the right-hand side of the underlying ODEs directly from trajectory data. On synthetic SIR, SEIR, and SEIRD data, FEX reports test mean squared errors on the order of $10^{-8}$ to $10^{-7}$, orders of magnitude lower than neural network and recurrent network baselines. On real COVID-19 data from Hubei, the method produces explicit formulas for recovered, deceased, and active cases using only three state variables and tracks the observed 15-day trajectory. The significance, if these results hold, is a data-driven route to interpretable and accurate epidemiological models for rapid public-health response.","feed_headline":"Symbolic regression recovers epidemic equations from data","feed_subtitle":"Reinforcement-learned expressions beat neural nets on synthetic SIR/SEIR/SEIRD and track 15-day COVID-19 forecasts.","key_machinery":"The engine is the Finite Expression Method: a library of unary and binary operators (such as $x$, $x^2$, $\\sin$, $\\exp$, $+$, $-$, $\\times$) is composed into binary expression trees, a reinforcement-learning controller generates operator sequences, and each sequence is scored by the inverse of its optimized loss, $S(e)=(1+L(e))^{-1}$. The policy is updated with a risk-seeking gradient to favor high-scoring expressions, and a candidate pool stores the best trees. The load-bearing identity for epidemiology is the per-component Euler residual loss (equation 3.5), which lets each coordinate of the vector field be learned independently from the same trajectory data; the authors use it with Euler integration and report that tree structures with one or two binary operators suffice for the tested models.","core_discovery":"The central claim is that a single symbolic-learning framework, FEX, can replace both manual model specification and black-box prediction in epidemiology. The paper demonstrates this by learning the full vector field of SIR, SEIR, and SEIRD models from simulated trajectories: each component $\\phi^{(i)}_{FEX}(x)$ is trained on the one-step Euler residual $\\|x^{(i)}_{s+1}-x^{(i)}_s-\\phi^{(i)}_{FEX}(x_s)\\Delta\\|^2$, and the discovered expressions reproduce the generating dynamics with test MSE between $10^{-8}$ and $10^{-7}$, compared with $10^{-4}$--$10^{-2}$ for a feedforward network and $10^{-6}$--$10^{-5}$ for an LSTM. On the real Hubei COVID-19 dataset, FEX fits the first 85 days and forecasts the next 15 days with explicit equations for $dR/dt$, $dD/dt$, and $dQ/dt$ that depend only on $R$, $D$, and $Q$, which the authors present as an interpretable alternative to the five-variable SEIQRDP model.","pith_inferences":["The Euler-residual training and the small operator library mean the recovered expressions are only as good as the assumption that the observed process is Markovian and noise-free at the daily reporting scale; adding realistic reporting noise or delays to the Hubei experiment would likely degrade the 15-day forecasts, which is a testable prediction of the paper's own framing.","The non-uniqueness acknowledged by the authors suggests that the specific Hubei formulas are not 'the' underlying law; a stronger validation would be to see whether FEX recovers the same functional form across different Chinese provinces or across re-sampled training windows.","Because the search is combinatorial, the method's cost scales with the operator library and tree depth; a natural extension is to seed the search with terms from known compartmental models (e.g., $SI$, $I^2$, $\\sin$ terms) to accelerate recovery on large systems.","If the accuracy on synthetic data transfers to real data with more compartments, FEX could be used to discover time-varying or spatially heterogeneous transmission rates, since the tree structure can represent products of state variables with coefficients."],"forward_implications":["FEX can serve as an automatic model-discovery tool: given incidence and outcome time series, it outputs closed-form ODEs without requiring an expert to pre-specify the compartment structure.","Because the learned equations are explicit, they can be inspected, compared across regions, and used to test intervention scenarios in ways black-box networks cannot.","If the reported accuracy holds, FEX-type symbolic models could replace or validate hand-tuned compartmental models in outbreak settings where expert models take too long to adapt.","The method's reliance on only three state variables for the Hubei data suggests that lower-dimensional descriptions may suffice for forecasting, reducing data requirements for future applications.","The same loss and search loop apply to any ODE system, so the approach is portable to other infectious diseases or other dynamical-systems forecasting tasks."],"supporting_citations":[{"why":"Introduces the FEX framework for solving high-dimensional PDEs; supplies the RL-based search loop and expression-tree construction used throughout.","marker":"[20]"},{"why":"Earlier work applying FEX to discover physical laws from data; establishes the method's general applicability that this paper extends to epidemiology.","marker":"[13]"},{"why":"Provides the risk-seeking policy-gradient objective used to update the controller toward high-scoring expressions.","marker":"[23]"},{"why":"Supplies the fractional-order SEIQRDP model and the Hubei COVID-19 dataset used as the real-world baseline and test case.","marker":"[2]"},{"why":"The Johns Hopkins/Our World in Data COVID-19 repository, the raw data source for the real-data experiments.","marker":"[14]"},{"why":"Classic SIR compartmental model used to generate one of the synthetic datasets.","marker":"[10]"},{"why":"SEIR model with vaccination, one of the synthetic test cases.","marker":"[1]"},{"why":"Provides the SEIRD compartmental model and notation used in the synthetic experiments.","marker":"[3]"}],"fun_headline_variants":["Finite Expression Method learns explicit epidemic equations from data","FEX symbolic learning uncovers epidemiological relationships with high accuracy","Reinforcement learning discovers explicit epidemic equations, beating LSTMs","FEX: interpretable symbolic models for outbreak forecasting","Symbolic regression with reinforcement learning models disease dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The accuracy and forecasting claims depend on the observed trajectories obeying the exact one-step Euler relation $x_{s+1}=x_s+\\phi(x_s)\\Delta$ with no measurement noise and with the same step size used in data generation or daily reporting; real epidemic data with reporting noise, delays, or hidden state variables would break this assumption.","fun_headline_variants_meta":{"raw":{"variants":["Finite Expression Method learns explicit epidemic equations from data","FEX symbolic learning uncovers epidemiological relationships with high accuracy","Reinforcement learning discovers explicit epidemic equations, beating LSTMs","FEX: interpretable symbolic models for outbreak forecasting","Symbolic regression with reinforcement learning models disease dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000604,"raw_usage":{"total_tokens":2792,"prompt_tokens":894,"completion_tokens":1898,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":510,"completion_tokens_details":{"reasoning_tokens":1820}},"tokens_in":510,"tokens_out":1898,"duration_ms":13677,"temperature":1.0,"reasoning_tokens":1820,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:03:19.457633+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the synthetic SIR/SEIR/SEIRD generators used in the paper, add Gaussian observation noise with standard deviation $10^{-3}$ to the training trajectories, and run FEX with the same settings; if the recovered expressions no longer approximate the true right-hand sides or test MSE rises above $10^{-4}$, the central accuracy claim holds only for noiseless data. For the real-data claim, retrain on the first 70 days of the Hubei series and forecast days 71--100; if the 30-day forecast diverges from reported counts much faster than the 15-day forecast, the learned equations are fitting short-term trends rather than the true dynamics.","supporting_citations":[{"cited_title":"Finite Expression Method for Solving High-Dimensional Partial Differential Equations","cited_arxiv_id":"2206.10121","evidence_quote":"Introduces the FEX framework for solving high-dimensional PDEs; supplies the RL-based search loop and expression-tree construction used throughout."},{"cited_title":"Finite expression methods for discovering physical laws from data","cited_arxiv_id":null,"evidence_quote":"Earlier work applying FEX to discover physical laws from data; establishes the method's general applicability that this paper extends to epidemiology."},{"cited_title":"Fractional-order seiqrdp model for simulating the dynamics of covid-19 epidemic","cited_arxiv_id":null,"evidence_quote":"Supplies the fractional-order SEIQRDP model and the Hubei COVID-19 dataset used as the real-world baseline and test case."},{"cited_title":"Covid- 19 data repository","cited_arxiv_id":null,"evidence_quote":"The Johns Hopkins/Our World in Data COVID-19 repository, the raw data source for the real-data experiments."},{"cited_title":"The mathematics of infectious diseases","cited_arxiv_id":null,"evidence_quote":"Classic SIR compartmental model used to generate one of the synthetic datasets."},{"cited_title":"Infectious diseases of humans: dynamics and control","cited_arxiv_id":null,"evidence_quote":"SEIR model with vaccination, one of the synthetic test cases."},{"cited_title":"Mathematical models in epidemiol- ogy, volume 32","cited_arxiv_id":null,"evidence_quote":"Provides the SEIRD compartmental model and notation used in the synthetic experiments."}],"review_version":1}