Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard $L^2$ regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. We introduce \textbf{OrthoReg} (Orthogonal Regularization), which directly penalizes overlap between the symbolic and neural components, preventing symbolic structure from being absorbed by the neural residual. This yields a complementary decomposition: the symbolic part captures what the library can express, and the neural part captures what remains. On benchmark dynamical systems with partial library mismatch, OrthoReg improves symbolic recovery and out-of-distribution behavior. [2606.19145] OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems Till Richter Affiliation:  Technical University of Munich Affiliation:  Helmholtz Munich Affiliation:  till.richter@helmholtz-munich.de Niki Kilbertus Affiliation:  Technical University of Munich Affiliation:  Helmholtz Munich Affiliation:  niki.kilbertus@helmholtz-munich.de Abstract Dynamical systems are fundamental to modeling the natural world, yet modeling them involves a persistent trade-off: manually prescribed mechanistic models are interpretable by design but often overly simplistic and misspecified; in contrast, flexible data-driven neural methods lack physical insight. Hybrid modeling aims for the best of both worlds by combining a prescribed or symbolic, physics-based component with a flexible neural network. A critical challenge, however, is that the neural component may relearn mechanistic parts, yielding redundant and uninterpretable models, especially when the symbolic structure itself is discovered from data. Existing methods based on standard L 2 L^{2} regularization rely on a projection argument that breaks when the symbolic component is learned through sparse discovery, allowing the neural augmentation to overlap with symbolic structure. Applications range from healthcare data  ( 7 ; 17 ; 39 ; 34 ) , climate modeling  ( 35 ; 12 ) , and power systems  ( 43 ) , just to name a few. However, it faces a fundamental trade-off: symbolic models, traditionally specified by hand, provide interpretability by design, but typically cannot capture complex unknown phenomena; flexible neural networks instead excel at fitting data from dynamical systems  ( 5 ) but lack physical insight. Hybrid modeling becomes essential when domain knowledge is partial: epidemiological models may capture core transmission dynamics but miss behavioral feedback mechanisms that modulate contact rates; mechanical systems follow known laws of motion but exhibit complex friction and damping effects not easily expressible in closed form; climate models encode fundamental physics but require data-driven corrections for sub-grid processes. In these scenarios, a purely symbolic approach underperforms due to missing phenomena, while purely neural models sacrifice the interpretability and physical consistency that domain experts require for scientific insight and decision-making. Hybrid modeling approaches  ( 32 ; 49 ; 50 ) combine physical priors (predetermined symbolic or parametrized expressions) with learned neural corrections expected to capture phenomena that are unknown or too complex to model directly. They promise the best of both worlds, but still require substantial prior knowledge in crafting the mechanistic part. This work focuses on sparse library-based discovery of the symbolic component, as in SINDy-style methods  ( 4 ) . In this regime, the symbolic component is fitted with a continuous sparsity penalty (e.g. L 1 L^{1} regularisation), where coefficient shrinkage leaves in-library residuals – the practically relevant case our analysis targets. Recent work improves symbolic discovery through physical constraints such as unit consistency ( 41 ) , transformer-based symbolic generation ( 25 ; 3 ; 45 ; 20 ; 46 ) , extensions to ODEs and trajectory data ( 2 ; 11 ; 40 ) , and methods for noisy, sparse, distributional, high-dimensional, or guided discovery settings ( 31 ; 9 ; 27 ; 42 ; 19 ) . However, when relevant effects are not representable by a compact symbolic library, purely symbolic models either fail or lose interpretability, motivating hybrid decompositions with a symbolic core and a flexible residual. Physics-informed neural networks. In the current formulation, one could simply set f aug ≡ f f_{\mathrm{aug}}\equiv f and f phy ≡ 0 f_{\mathrm{phy}}\equiv 0 . However, this would undermine the entire idea of hybrid modeling. When the physical model class is fixed, 49 provide thorough theoretical guarantees showing that a relatively simple norm-based regularization scheme is sufficient to ensure that f aug f_{\mathrm{aug}} “only captures what is necessary, but not more.” A simplified vector-field version of this norm-regularized principle is min f phy ∈ ℱ phy , f aug ∈ ℱ ⁡ ‖ f − f phy − f aug ‖ 𝒟 2 + λ 2 ​ ‖ f aug ‖ 𝒟 2 . Advances in Neural Information Processing Systems 36 , pp. 12929–12950 . Cited by: §2 . [48] P. Wyder, J. Goldfeder, A. Yermakov, Y. Zhao, S. Riva, J. Williams, D. Zoro, A. Rude, M. Tomasetto, J. Germany, et al. (2026) Common task framework for a critical

The synergistic control of multiple pollutants is critically challenged by complex nonlinear interactions, strong spatiotemporal heterogeneity, and the difficulty of tracing causal drivers. Deep learning offers high predictive power but suffers from the "black-box" problem, limiting its acceptance in environmental decision-making. Explainable Deep Learning (XDL) integrates physical mechanisms with interpretable algorithms, achieving both prediction accuracy and explanatory transparency. This review systematically evaluates the effectiveness and limitations of XDL in analyzing multi-pollutant interactions, with a comparative focus on atmospheric and aquatic environments. Key techniques, including SHAP, attention mechanisms, and physics-informed neural networks, are examined for their roles in synergistic monitoring, source apportionment, and regulatory optimization. The main findings reveal that: (1) XDL, particularly the "tree model + SHAP" paradigm, has become a dominant tool for quantifying driving factors, yet most attributions remain correlational rather than causal; (2) physics-informed fusion (soft vs. hard constraints) improves physical consistency but faces unresolved conflicts between data and physical laws, with current models lacking a conflict detection mechanism; (3) cross-media comparison shows a unified technical logic of "physical mechanism guidance + post hoc feature attribution", but atmospheric applications lead in embedding advection-diffusion constraints, while aquatic research excels in spatial topology modeling via graph neural networks; (4) critical bottlenecks include the lack of causal inference, uncertainty-unaware interpretations, and data scarcity. Future directions demand a shift from correlation-only to causal-aware attribution, from blind fusion to conflict-detecting systems, and from no evaluation standards to domain-specific validation benchmarks. XDL is poised to transform multi-pollutant governance from experience-driven to intell Limitations of Traditional Control Technologies XDL, with its ternary structure of “deep learning architecture + explainable module + physical constraint”, has initially demonstrated the potential to meet the technical requirements for synergistic control of multi-pollutants. However, physicochemical models—built upon fundamental laws of fluid dynamics and thermodynamics, such as the WRF-Chem model for atmospheric applications and the EFDC model for water bodies—rely heavily on comprehensive emission inventories and parameter settings. A more fundamental issue is that current XDL lacks a physical conflict detection mechanism: when predictions deviate significantly from physical residual terms, the model cannot distinguish between data anomalies, erroneous equations, or missing processes, nor can it provide actionable warnings to decision-makers. Breaking through these bottlenecks requires reconstructing the physics-fusion paradigm of XDL. A study on PM 2.5 estimation in China explicitly pointed out that when temperature is highly correlated with temporal encoding, one-dimensional partial dependence plots produce a spurious relationship of “rising temperature leading to increased PM 2.5 ” that contradicts physical reality, whereas two-dimensional partial dependence plots reveal the true negative correlation [ 69 ]. Second, most studies only report feature importance rankings, with few exploring the attribution stability of models across different concentration intervals (e.g., light and heavy pollution). To address this, some studies have attempted to compensate for this deficiency by incorporating physical constraints, such as introducing residual terms of the continuity equation into the loss function to ensure that deep learning model predictions comply with the law of mass conservation [ 68 ]. This physics-informed fusion approach provides a feasible path from statistical correlation to causal inference, but its integration into the SHAP interpretation framework is still in its infancy. 5.2. Research in the Laoshan River Basin shows that the contribution of turbidity (TU) to TP prediction rises significantly in the wet season due to enhanced non-point source pollution input via runoff, yet this seasonal pattern is nearly flattened in annual average SHAP values [ 77 ]. A more fundamental issue is that the attribution provided by SHAP is essentially “correlational” rather than “causal”. Although researchers often interpret high SHAP values as “driving factors”, the model only captures statistical correlations. While this “process–data fusion” approach is directionally correct, it essentially remains an improvement at the feature engineering level and has not yet achieved a methodological leap from statistical correlation to causal inference. 6.2. Physics-Informed and Multi-Task Learning Frameworks for Synergistic Prediction of Multiple Pollutants To address the limitation of single-pollutant modeling that overlooks interactions among pollutants, multi-task learning and physics-informed fusion models have emerged as a new research focus. To enable cross-jurisdictional regulatory acceptance, future standards must require that data augmentation methods pass physical sanity checks—for example, no generation of “high pesticide with zero rainfall” samples—otherwise interpretations may be legally challengeable. 9. Conclusions XDL advances multi-pollutant synergistic control by enabling nonlinear attribution and physics-informed learning. Yet a deeper, often overlooked problem remains: interpretability lacks a verifiable ground truth. SHAP provides statistical correlations; physics constraints enforce mass conservation or advection–diffusion laws. When they disagree—e.g., monitoring data show a concentration surge but physical equations predict a decline—current XDL has no mechanism to detect, quantify, or signal the conflict. Consequently, users receive “explanations” that may be physically implausible without any warning.