Title: Discovering Symbolic Partial Differential Equation by Abductive Learning
Venue: NeurIPS 2025
Accepted: yes
Keywords: Abductive Learning

AI4Science
Abstract: Discovering symbolic Partial Differential Equation (PDE) from data is one of the most promising directions of modern scientific discovery.
    Effectively constructing an expressive yet concise hypothesis space and accurately evaluating expression values, however, remain challenging due to the exponential explosion with the spatial dimension and the noise in the measurements.
    To address these challenges, we propose the ABL-PDE approach that employs the Abductive Learning (ABL) framework to discover symbolic PDEs.
    By introducing a First-Order Logic (FOL) knowledge base, ABL-PDE can represent various PDEs, significantly constraining the hypothesis space without sacrificing expressive power, while also facilitating the incorporation of problem-specific knowledge.
    The proposed consistency optimization process establishes a synergistic interaction between the knowledge base and the neural network learning module, achieving robust structure identification, accurate coefficient estimation, and enhanced stability against hyperparameter variation.
    Experimental results on three benchmarks across different noise levels demonstrate the effectiveness of our approach in PDE discovery.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents a methodology for discovering the differential operator of Partial Differential Equations using Abductive Learning. The method is structured in two steps, first a neural network is used to approximate the given data such that there is no need for an integrator and second a process that contains an outer and an inner loops is considered. The outer loop increases the length of the differential operator, the inner loop picks the candidate operators, use Mean Subset Selection to find the shorted expression and fine-tune the neural network. The process is performed until convergence on a level margin criterion. The method is tested on three PDEs for different levels of noise against different baselines, and on two PDEs specifically against D-CIPHER which cannot handle systems.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n- This is an interesting method that combines canonical grammars that are expressive enough, see Theorem 1, together with a consistency optimization where is showed that the k-level margin decreases monotonically. The method essentially proposes the idea of reducing the hypothesis using a first-order logic knowledge base and integrity constraints, and delivers state-of-the-art accuracy for PDEs with different levels of noise. \n\nOverall, it provides a novel method to cut down the hypothesis space of combinatorial optimization methods, which is very valuable. Also the paper is mostly well structured, well written, and the authors provide helpful information in the appendix. \n\nWeaknesses:\n\nClarity: I believe that there are some parts that are not completely clear to me. For example:\n\n- How are constants treated in the Grammar. The non-terminal <unit> allows for (in-)dependent variables, and partials only, meaning that constants, for example a source term \"+2\", cannot be generated. However, constants appear in the target equations, for example 0.01/pi in the Burger's. In my understanding, the authors rely on the \\xi vector in the FEM training to account for that but this doesn't work for additive constants, such as u_t - D u_xx - S where S a constant source. \n\n- There is the following subtle issue in the method (at least to my understanding): The strength of using a grammar is to keep the library of candidate term small such that the MSS choice remains tractable. Theorem 1 states that any expression, even if it is nested, can be re-written as linear combination of terms from the canonical grammar. However, when this happens a single complex node is substituted by multiple simpler nodes, which means that you might lose the benefit of restricting the space of candidate expressions and having a large library again. \n\n- I believe that there is one more subtle issue with Theorem 1. In my understanding, Theorem 1 guarantees equivalence of symbolic expressions, but does not guarantee that those equivalences hold as PDE operators acting on a solution space with the correct boundary conditions. The difference is that without discussing boundary condition assumptions, probably smoothness also because maybe the solution has weak derivatives, you can be sure that the expansion holds in the weak form where you consider in the FEM residual. \n\n- In my understanding, Theorem 1 assumes that derivatives and variables commute and the grammar contains the basic terms, but it seems like constants are missing from the grammar. \n\n- It is not clear how the FEM based training handles boundary conditions in the weak formulation. It seems like the training ignores what happens on the boundary. Moreover, I believe that there should be some discussion on how the conditioning of the mass matrix affects the overall  training, as M might be ill-conditioned for coarse meshes. \n\n- It is not clear to me how expensive and numerically stable is the LevelMargin step as it needs to compute orthogonal projections which is expensive and might be numerically unstable for colinear terms.\n\nLemma 5:\n\n- In the proof it is stated that \"Without a loss of generality we assume $v \\perp Y$.\". This statement is not justified and the derivations that follow depend critically on that orthogonality. This statement is not without a loss of generality because if v has a component inside Y, replacing by its orthogonal component v_{ortho} = v - Proj_Y v changes the left-hand side of the inequality because the projection onto span(v) does not coincide with the projection on span(v_{ortho}).  This means that the inequality in Theorem 2 does not necessarily hold\n\n. If the authors impose the condition $v \\perp Y$, the inequality is true but the method does not apply to arbitrary grammars, because you would need to guarantee the orthogonality. \n\n- The results for D-CIPHER are (in my understanding) in contradiction with the original paper, that shows that D-CIPHER is robust to noise.\n\nQUESTIONS:\n- Could you provide more information about how the method handles additive coefficients?\n\n- When performing re-writes of complex terms in the grammar, is there a way to derive a bound to this growth such the method remains efficient when considering more complex expressions (ones with forcing terms for example).\n\n-  Maybe I missed it but could you clarify if the candidate terms are the same for all PDEs or not? And if not how they are chosen?\n\n- How many times did you run D-CIPHER and what was the best result?\n\n- Why is the discovery more accurate for 5% error?\n\nLIMITATIONS:\nThere is only one limitation stated in the conclusions. I believe that the authors should include bullets from the weaknesses section to the limitations.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper introduces ABL-PDE, a novel abductive learning-based approach for PDE discovery. The key contribution is the transformation of the traditional unidirectional discovery pipeline into a bidirectional enhancement loop, improving both structure and parameter identification. Specifically, the method incorporates FOL KB to represent a broad class of PDEs with reduced redundancy and enhanced interpretability. Extensive experiments across multiple PDE tasks and noise levels demonstrate that ABL-PDE achieves state-of-the-art performance in both equation structure recovery and coefficient estimation.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n1. The authors replaces the traditional unidirectional pipeline with a feedback loop between symbolic and numerical components, enhancing robustness and accuracy.\n2. The authors introduce a knowledge base to constrain the candidate search space, effectively reducing the problem size to a computationally manageable scale. This approach is both practical and innovative - I like this idea.\n\n### Weaknesses\n1. The Experiments section includes a limited number of baseline methods for comparison. The method could benefit from comparisons with more recent or diverse PDE discovery methods.\n2. The alternating optimization between MSS and FEMTrain lacks a theoretical guarantee of convergence, which may affect reproducibility or stability in some scenarios.\n3. The choice of using L2RE of term coefficients can not clearly reflect the performance of equation discovery.\n\nQUESTIONS:\n1. In Table 2, the authors compare the performance of ABL-PDE and its variants with the baselines PDERidge and DL-PDE++. I have a few questions and comments:\n(1) I noticed that some L2RE values are exactly the same across different methods for the same PDE and noise level. Is this because the same Ridge regression is used in the final step of those methods?\n(2) Why are standard deviations (± values) reported for some results but not others? Is this because those experiments are deterministic, with no variation across runs?\n(3) I understand that the dagger symbol indicates the discovered PDE includes redundant terms. This is common in symbolic regression. However, how severe are these cases? For example, one extra term with a tiny coefficient might be fine, but having five extra terms with meaningful coefficients in a three-term PDE could be considered a failure. Also, does the dagger indicate that all correct terms are present and only redundant ones are added, with nothing missing?\n(4) If an extra term is included in the discovered PDE, how does the L2RE be calculated? Do you ignore the coefficients of redundant terms and only use the correct terms when computing the error?\n\n2. Could the authors compare their method and its variants with more recent and representative baselines? PDERidge only applies Ridge regression on a known PDE structure, which is not a complete PDE discovery method. DL-PDE++ (2021) is somewhat outdated, though still acceptable. It would be helpful to include comparisons with more modern methods such as Symbolic Physics Learner (Sun et al., ICLR 2023) or SGA-PDE (Chen et al., Physical Review Research 2022).\n\n3. Since the overall optimization process alternates between MSS and FEMTrain, can the authors provide a proof or argument that this process always converges?\n\n4. The caption of Figure 3 mentions that 5 different values of lambda (used in Equation 2) are considered. However, neither Figure 3 nor Appendix G clearly shows or discusses these different lambda values. Could the authors clarify this?\n\n5. In section 6.4, it is stated that the lambda values used are [0.5λ, 0.75λ, λ, 1.25λ, 1.5λ]. Does this mean five separate configurations with λ set to 0.5, 0.75, 1.0, 1.25, and 1.5 respectively? The wording is a bit unclear.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 4, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

comments: SUMMARY:
The idea of this paper is to integrate a neural network for fitting the solution field (and estimating derivatives) with a first-order logic (FOL) knowledge base that generates canonical expression terms via abductive logic programming.

STRENGTHS AND WEAKNESSES:
Strength:  Combining FOL abductive learning to systematically constrain symbolic PDE search is quite original.
Weaknesses
While the canonical forms reduce combinatorial explosion, the abductive reasoning and MSS steps could still become heavy for large systems. It would help to see timings or scaling on more complex multi-physics PDEs or with >2 dependent variables.
Though standard PDEs were used, the inclusion of a more realistic engineering problem (e.g., turbulent flow, reaction-diffusion with unknown kinetics) could strengthen the practical impact.

QUESTIONS:
can you add add a synthetic example where the underlying PDE has unknown nonlinear parameter functions to see how the current framework fails or could be extended.

LIMITATIONS:
They test only on well-known equations (Burgers, Schrödinger, Navier-Stokes in 2D).
If the initial pre-training fails (very sparse or chaotic data), the abductive loop may not recover.
TITLE: Official Review
CLARITY: 3
IMPACT: 3
ORIGINALITY: 3
DATE: 2025-07-02

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors present a new approach for automatic discovery of partial differential equations from data using neural network learning and first order logic based reasoning. Current approaches for partial equation discovery suffers from several issues including - large hypothesis space which makes it challenging to search for the correct equation, inaccurate derivative estimation during expression evaluation as only discrete measurements are available. To address these issues, the authors propose to use first-order-logic-based reasoning to constrain the hypothesis space of possible expressions/equations, and neural network based derivative estimation for expression evaluation. The authors’ framework also contains an optimization module to ensure consistency between these two modules. Experimental results are provided to demonstrate the efficacy of the proposed framework.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\nThe authors presented a good introduction and motivation for their work. Related works are discussed along with their limitations. Algorithm for Monotone subset selection is presented in detail. Theoretical analysis is presented. Experimental results are organized with key questions mentioned at the beginning. Ablation study is provided to show the importance of different modules of the framework.\n\nWeakness:\n1. The authors may consider adding the following details to their manuscript:\n2. Why is it necessary to have a consistency maximization process? There can be one/two sentences between line 54-62 describing the intuition behind it.\n3. In the contribution, the authors mention bidirectional loop, however symbolic regressions method like genetic programming also has this kind of iterative loop structure, how is the authors’ method different from that?\n4. What is the input/output of the neural network used for derivative estimation? There are many methods for approximating derivatives and the authors mention their limitations in the introduction. How does having a neural network mitigate those issues?\n5. Can the authors give an example of how abduction works as indicated in section 4.1, step 2. It is currently hard to understand how abduction relates the number of operators to expand the library of candidates.\n6. Table 1 integrity constraints are hard to follow at present, despite the authors giving details. Can you please elaborate more on one constraint, maybe C1 as to why this will give a unique order.\n\nQUESTIONS:\nMentioned in the strength and weakness section\n\nLIMITATIONS:\nPlease see the strength and weakness section", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-04"}