Title: PINNs with Learnable Quadrature
Venue: NeurIPS 2025
Accepted: yes
Keywords: PINNs
Abstract: The growing body of work on Physics-Informed Neural Networks (PINNs) seeks to use machine learning strategies to improve methods for solution discovery of Partial Differential Equations (PDEs). While classical solvers may remain the preferred tool of choice in the short-term, PINNs can be viewed as complementary. The expectation is that in some specific use cases, they can be effective, standalone. A key step in training PINNs is selecting domain points for loss evaluation, where Monte Carlo sampling remains the dominant but often suboptimal in low dimension settings, common in physics. We leverage recent advances in asymptotic expansions of quadrature nodes and weights (for weight functions belonging to the modified Gauss-Jacobi family) together with suitable adjustments for parameterization towards a data-driven framework for learnable quadrature rules. A direct benefit is a performance improvement of PINNs, relative to existing alternatives, on a wide range of problems studied in the literature. Beyond finding a standard solution for an instance of a single PDE, our construction enables learning rules to predict solutions for a given family of PDEs via hyper-networks, a useful capability for PINNs.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an enhancement to physics-informed neural networks (PINNs) by introducing learnable quadrature rules. Specifically, the weight function in the quadrature rule is parameterized by a neural network. The loss function includes regularization to discourage small weights and enforces that the learned quadrature weights sum to the integral of the weight function. The method shows improved empirical performance on several 1D benchmark problem. The authors also extend their methods to train neural operators without using numerical solutions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths\n- The method is compared with lots of existing methods and shows empirical success.\n- The approach is well motivated and is principled in adaptive quadrature.\n\nWeaknesses\n- All examples are 1D in space.\n- The presentation lacks clarity in several aspects. The use of $\\theta$ and $\\phi$ is confusing. For example, $w_\\theta$ in Equation (2) and (10), $w_\\phi$, $u_\\theta$ in Algorithm 1, and $\\phi$ goes to the solution function in Figure 4. In addition, in Algorithm 1, why is $\\phi(\\mu)$.\n\nQUESTIONS:\n- How does the learned weight function compare to classical choices such as Gauss-Jacobi weight functions? Please provide plots or examples of the learned weight functions for illustration.\n\n- Equation (30), which defines $l_w$, is unclear. It is not evident which parameters are trainable. Are the quadrature weights $w_i$ themselves learnable? It seems that for any given weight function $w(x)$, the weights $w_i$ and nodes $x_i$ are computed using a numerical procedure described in Section 5.\n\n- The paragraph on the interlacing of roots of orthogonal polynomials is confusing. What exactly does “overfitting” mean in this context? A plausible failure mode would be the learning of a trivial weight function (e.g., $w(x) = 0$), which seems to motivate the penalty term in Equation (20). However, the paper states: “By utilizing quadrature nodes and weights stemming from varying degrees of orthogonal polynomials (all of which correspond to the same weight function being learned), we introduce the desired stochasticity to prevent over-fitting.” This seems to be a key component of the algorithm, but it is unclear how it is implemented or integrated into the training process.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-18"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a principled method for learning adaptive quadrature points and weights, which improves the accuracy of PINN. The quadrature point is parameterized as the roots of a parameterized Jacobi polynomial induced from a modified Gauss-Jacobi type weight function, where a learnable multiplicative factor $h_{\\theta}(x)$ is introduced. To compute the roots efficiently, the authors utilize the recent advances in asymptotic expansion of OP described in section 5.1.  Since the quadrature points are implicit functions of the parameters $\\theta$ in $h_{\\theta}(x)$, the author uses implicit differentiation over the simplified root equation (eq. 17-19). Learning is done by backpropagating the PINN loss over the parameters of the solution function and the parameters of the quadrature point. The author also proposed an additional loss for the quadrature point in equation 20. The method was tested on various common PDEs and demonstrated superior accuracy (measured in L2 relative error) against other methods. I believe that this work made a great contribution in improving PINN and making it closer to being comparable to the existing PDE solvers.\n\nSTRENGTHS AND WEAKNESSES:\nStrength:\n1. The method is very principled and demonstrates strong empirical performance\n2. The writing was very clear, and the paper is easy to follow despite the heavy mathematical content\n3. To the best of my knowledge, the method is novel, as previous methods for learning quadrature points do not generate the quadrature points as roots of OP.\n\nWeaknesses:\n1. There should be a comprehensive description of the parameterization. Also, Algorithm 1 provides a sketch of the training process; the bulk of the work in this paper is in how the quadrature points are computed. In this current form, I had to gather various pieces from the paper to understand how exactly the quadrature points are computed. Also, I couldn't find the network architecture description\n2. The method's accuracy should also be benchmarked against non-PINN PDE solvers. It is good to show that the method outperforms other PINN variants, but it is also good to see how much gap there still is.\n3. The method is only tested on low-dimensional standard PDEs. Applicability to real-world problems could be challenging, especially for irregular domains\n\nQUESTIONS:\n1. $\\alpha$ and $\\beta$ in the weight function are not learned. How are they chosen? Same question for the expansion coefficients. What kind of information about the equation do we need to derive these?\n2. To derive equation 18, you assume the roots are real and there are only simple poles. When does this condition hold, and could this assumption be violated in some PDEs?\n3. It appears that the first term in the quadrature loss in equation 20 is a relaxation of a hard constraint. How much is this error? And what effects does this error have on the solution?\n4. You mentioned that the quadrature point could be learned adversarially, but in practice, no significant difference was observed. What could be the reason?\n5. In line 230, it was mentioned that \"we use linear layers...\", but from the appendix, it seems that the author is actually using a multi-layer perception with 3~4 layers. I fail to understand this paragraph.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-27"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors propose \"LearnQuad\", an approach to sample quadrature points for physics-informed neural network training (PINN training).\nThey demonstrate the performance of the approach in comparison to other sampling strategies in a simple setting (Diffusion equation), and separately demonstrate that quadrature points can also be found for families of PDEs.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths: \n1) Several numerical experiments for low-dimensional PDEs, and comparison to other sampling techniques.\n2) Appendix with description of all PDEs and settings.\n\nWeaknesses:\nThe paper introduces a method to sample quadrature points to improve PINN training. In general, improving methods that are still sub-par compared to the state of the art (here: compared to all PDE solvers) is fine, but the comparison should still make it clear how far away the current method still is from the SOTA. This has been discussed extensively in the PINN literature at this point, see for example:\n * McGreivy, Nick, and Ammar Hakim. \"Weak Baselines and Reporting Biases Lead to Overoptimism in Machine Learning for Fluid-Related Partial Differential Equations.\" Nature Machine Intelligence 6, no. 10 (September 25, 2024): 1256–69. https://doi.org/10.1038/s42256-024-00897-5.\n * Wang, Sifan, Yujun Teng, and Paris Perdikaris. \"Understanding and Mitigating Gradient Flow Pathologies in Physics-Informed Neural Networks.\" SIAM Journal on Scientific Computing 43, no. 5 (January 2021): A3055–81. https://doi.org/10.1137/20M1318043.\n * Jorge F. Urbán. \"Unveiling the Optimization Process of Physics Informed Neural Networks: How Accurate and Competitive Can PINNs Be?\" Journal of Computational Physics 523 (February 15, 2025): 113656. https://doi.org/10.1016/j.jcp.2024.113656.\n\nThe first citation includes important suggestions to developing new methods for PINNs, the second and third are important contributions to understanding and improving PINNs. Especially the third demonstrates that the accuracy can be drastically improved by choosing a different optimizer. These developments have to be taken into account when developing new methods for PINNs, otherwise it is impossible to understand how well the new method performs in practice.\n\n1) Weak baselines.\n2) Comparison to classical solvers / solvers in latest PINN literature is missing; points in appendix 11.3 are not enough to address this appropriately.\n3) No theoretical analysis of the approach. Stating that \"no additional explicit error estimation is required\" is not enough to convince the reader why the proposed method is providing \"better\" samples than monte carlo approximation.\n4) A lot of formatting concerns (see below).\n5) The only comparison to other methods is done on a very simple benchmark (diffusion equation 1d) with only 30 sample points. It is impossible to judge if the method performs better than others (especially classical methods) with just this experiment.\n\nQUESTIONS:\nAlso see \"weaknesses\".\nQ1) how would the method perform when using the best PINN optimizers instead of Adam?\nQ2) how accurate are classical methods for the given PDEs, with the same number of sample points?\nQ3) how do the other sampling approaches perform when the number of points increases, or when the PDE is different?\n\nLIMITATIONS:\nThe limitations section does not describe a limitation of LearnQuad, but of PINNs in general. No limitations of the present work are discussed.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper combines the numerical approach with the machine learning approach to learn solutions of PDEs. It proposes the method LearnQuad that enhances the physics informed neural networks (PINNs) with learnable quadrature rules, that are data-driven learned and parametrized by neural networks. By using asymptotic expansions, the method efficiently obtains quadrature nodes and weights from the learned weight function. This method can be applied in both the strong form and weak form of solving PDEs. Additionally, compared to operator learning, this method is able to generalize to solving a family of PDEs, and it does not need solution and PDE parameter pairs. For solving a single PDE, this method yields improved performance over multiple baselines including PINN, CPINN, RAD, etc. For solving a family of PDEs, this method is well generalized to the test set.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. For choosing collocation points and their weights, several prior works have been proposed. This method proposes a data-driven method to dynamically learn collocation points and weights. Unlike fixed quadrature or random sampling, the learned quadrature rules is tailored to the structure of the PDEs, enabling a more effective use of limited collocation points. \n\n2. This paper bridges the gap between PINNs and neural operators for solving a family of neural operators. However, the proposed method doesn’t need the use of labelled pairs of solutions and PDE parameters.\n\nWeaknesses:\n1. The biggest weakness is that experiments use 1000 collocation points. Though this paper claims the method is efficient in obtaining a large number of collocation points and weights by leveraging the asymptotic expansions, it is unclear the performance gap would still exist by gradually increasing the number of collocation points. It is also unclear if the asymptotic expansions would be numerically stable at larger scales. \n\n2. Second, the PDEs solutions used in the experiments are smooth. It concerns me that the method based on orthogonal polynomials might be less effective for problems with PDEs solutions that are more complex, e.g., solutions that have multi-scale and are more chaotic. Therefore, it is unclear if the proposed method would keep its accuracy advantage in those settings. It is interesting to see if with 1000 collocation points or by gradually increasing the number of points, how would the proposed method perform compared to baselines.\n\nQUESTIONS:\n1. Is it possible for the authors to show the total training time for all methods? Since LearnQuad introduces additional components, it is important to quantify the computational cost compared to PINNs or other baselines. I am also curious how the number of collocation points would scale the training time.\n\n2. It is unclear how the \\alpha and \\beta in equation (10) is chosen. How much effort do the authors put on for tuning the two parameters? How do different choices of \\alpha and \\beta affect the results? An ablation study and analysis would be appreciated.\n\n3. How does the method work in the high dimensional PDE problem in terms of the accuracy and computational cost? Is there any optimization challenge or training instability issue for the learnable quadrature rules?\n\n4. For learning a family of PDEs, the proposed method can be used without labelled data, which is needed for neural operators. I am curious how neural operators perform in the test set. I think the authors can train neural operators either with the solutions learned by the proposed model or with the ground truth solutions.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}