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  • Open Access

Physics-informed neural networks for solving derivative-constrained partial differential equations

Kentaro Hoshisashi*, Carolyn E. Phelan, and Paolo Barucca

  • *Contact author: k.hoshisashi@ucl.ac.uk
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Phys. Rev. E 113, 055301 – Published 4 May, 2026

DOI: https://doi.org/10.1103/5bbf-p6zk

Abstract

Physics-informed neural networks (PINNs) recast partial differential equations (PDEs) solving as an optimization problem in function space by minimizing a residual-based objective, yet many applications require additional derivative-based relations that are just as fundamental as the governing equations. In this paper, we present derivative-constrained PINNs (DC-PINNs), a general framework that treats constrained PDE solving as an optimization guided by a minimum objective function criterion where the physics resides in the minimum principle. DC-PINNs embed general nonlinear constraints on states and derivatives, e.g., bounds, monotonicity, convexity, incompressibility, computed efficiently via automatic differentiation, and they employ self-adaptive loss balancing to tune the influence of each objective, reducing reliance on manual hyperparameters and problem-specific architectures. DC-PINNs consistently reduce constraint violations and improve physical fidelity versus baseline PINN variants, representative hard-constraint formulations on benchmarks, including heat diffusion with bounds, financial volatilities with arbitrage-free, and fluid flow with vortices shed. Explicitly encoding derivative constraints stabilizes training and steers optimization toward physically admissible minima even when the PDE residual alone is small, providing reliable solutions of constrained PDEs grounded in energy minimum principles.

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References (47)

  1. M. Raissi, P. Perdikaris, and G. E. Karniadakis, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, J. Comput. Phys. 378, 686 (2019).
  2. J. Sirignano and K. Spiliopoulos, DGM: A deep learning algorithm for solving partial differential equations, J. Comput. Phys. 375, 1339 (2018).
  3. J. Berg and K. Nyström, A unified deep artificial neural network approach to partial differential equations in complex geometries, Neurocomputing 317, 28 (2018).
  4. W. E and B. Yu, The Deep Ritz method A deep learning-based numerical algorithm for solving variational problems, Commun. Math. Stat. 6, 1 (2018).
  5. E. Kharazmi, Z. Zhang, and G. E. Karniadakis, hp-VPINNs: Variational physics-informed neural networks with domain decomposition, Comput. Methods Appl. Mech. Eng. 374, 113547 (2021).
  6. L. Onsager, Reciprocal relations in irreversible processes. I., Phys. Rev. 37, 405 (1931).
  7. Y. Patel, V. Mons, O. Marquet, and G. Rigas, Turbulence model augmented physics-informed neural networks for mean-flow reconstruction, Phys. Rev. Fluids 9, 034605 (2024).
  8. G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang, Physics-informed machine learning, Nat. Rev. Phys. 3, 422 (2021).
  9. Z. Chen, Y. Liu, and H. Sun, Physics-informed learning of governing equations from scarce data, Nat. Commun. 12, 6136 (2021).
  10. I. E. Lagaris, A. Likas, and D. I. Fotiadis, Artificial neural networks for solving ordinary and partial differential equations, IEEE Trans. Neural Networks 9, 987 (1998).
  11. A. S. Krishnapriyan, A. Gholami, S. Zhe, R. M. Kirby, and M. W. Mahoney, Characterizing possible failure modes in physics-informed neural networks, Adv. Neural Inf. Process. Syst. 34, 26548 (2021).
  12. S. Cuomo, V. S. Di Cola, F. Giampaolo, G. Rozza, M. Raissi, and F. Piccialli, Scientific machine learning through physics–informed neural networks: Where we are and what's next, J. Sci. Comput. 92, 88 (2022).
  13. S. Wang, S. Sankaran, and P. Perdikaris, Respecting causality for training physics-informed neural networks, Comput. Methods Appl. Mech. Eng. 421, 116813 (2024).
  14. S. Wang, X. Yu, and P. Perdikaris, When and why PINNs fail to train: A neural tangent kernel perspective, J. Comput. Phys. 449, 110768 (2022).
  15. A. D. Jagtap, E. Kharazmi, and G. E. Karniadakis, Conservative physics-informed neural networks on discrete domains for conservation laws: Applications to forward and inverse problems, Comput. Methods Appl. Mech. Eng. 365, 113028 (2020).
  16. K. Lo and D. Huang, On training derivative-constrained neural networks, arXiv:2310.01649.
  17. N. Sukumar and A. Srivastava, Exact imposition of boundary conditions with distance functions in physics-informed deep neural networks, Comput. Methods Appl. Mech. Eng. 389, 114333 (2022).
  18. L. Lu, R. Pestourie, W. Yao, Z. Wang, F. Verdugo, and S. G. Johnson, Physics-informed neural networks with hard constraints for inverse design, SIAM J. Sci. Comput. 43, B1105 (2021).
  19. Y. Chen, D. Huang, D. Zhang, J. Zeng, N. Wang, H. Zhang, and J. Yan, Theory-guided hard constraint projection (HCP): A knowledge-based data-driven scientific machine learning method, J. Comput. Phys. 445, 110624 (2021).
  20. T. Beucler, M. Pritchard, S. Rasp, J. Ott, P. Baldi, and P. Gentine, Enforcing analytic constraints in neural networks emulating physical systems, Phys. Rev. Lett. 126, 098302 (2021).
  21. L. D. McClenny and U. M. Braga-Neto, Self-adaptive physics-informed neural networks, J. Comput. Phys. 474, 111722 (2023).
  22. S. Wang, S. Sankaran, H. Wang, and P. Perdikaris, An expert's guide to training physics-informed neural networks, arXiv:2308.08468.
  23. H. Son, S. W. Cho, and H. J. Hwang, Enhanced physics-informed neural networks with Augmented Lagrangian relaxation method (AL-PINNs), Neurocomputing 548, 126424 (2023).
  24. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, New York, 2006).
  25. D. P. Bertsekas, Constrained Optimization and Lagrange Multiplier Methods (Academic Press, New York, 1982).
  26. X. Glorot and Y. Bengio, Understanding the difficulty of training deep feedforward neural networks, in Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, edited by Y. W. Teh and M. Titterington, Proceedings of Machine Learning Research Vol. 9 (PMLR, Chia Laguna Resort, Sardinia, Italy, 2010), pp. 249–256.
  27. D. P. Kingma and J. Ba, Adam A method for stochastic optimization, in Proceedings of the 3rd International Conference on Learning Representations (ICLR, San Diego, 2015)
  28. J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, Y. Katariya, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX Composable transformations of Python+NumPy programs, version 0.3.13, http://github.com/jax-ml/jax (2018).
  29. J. Heek, A. Levskaya, A. Oliver, M. Ritter, B. Rondepierre, A. Steiner, and M. van Zee, Flax A neural network library and ecosystem for JAX, version 0.12.7, http://github.com/google/flax (2024).
  30. Google, https://research.google.com/colaboratory/faq.html (accessed on 25th Sept, 2025).
  31. J. R. Cannon, The One-Dimensional Heat Equation, Encyclopedia of Mathematics and its Applications (Cambridge University Press, Cambridge, 1984).
  32. B. Dupire, Pricing with a smile, Risk 7, 18 (1994).
  33. S. Kim, H. Han, H. Jang, D. Jeong, C. Lee, W. Lee, and J. Kim, Reconstruction of the local volatility function using the Black–Scholes model, J. Comput. Sci. 51, 101341 (2021).
  34. P. P. Boyle and D. Thangaraj, Volatility estimation from observed option prices, Decisions Economics Finance 23, 31 (2000).
  35. R. Peyret and T. D. Taylor, Computational methods for fluid flow, Springer Series in Computational Physics (Springer-Verlag, New York, 1983).
  36. G. K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 2000)
  37. D. Moxey, C. D. Cantwell, Y. Bao, A. Cassinelli, G. Castiglioni, S. Chun, E. Juda, E. Kazemi, K. Lackhove, J. Marcon, et al., Nektar++: Enhancing the capability and application of high-fidelity spectral/ hp element methods, Comput. Phys. Commun. 249, 107110 (2020).
  38. M. D. McKay, R. J. Beckman, and W. J. Conover, A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 42, 55 (2000).
  39. Z. Gao, L. Yan, and T. Zhou, Failure-informed adaptive sampling for PINNs, SIAM J. Sci. Comput. 45, A1971 (2023).
  40. L. Lu, X. Meng, Z. Mao, and G. E. Karniadakis, DeepXDE: A deep learning library for solving differential equations, SIAM Rev. 63, 208 (2021).
  41. R. Frostig, M. J. Johnson, and C. Leary, Compiling machine learning programs via high-level tracing, in Proceedings of SysML 2018 (Stanford, CA, USA, 2018).
  42. A. D. Jagtap and G. E. Karniadakis, Extended physics-informed neural networks (XPINNs): A generalized space-time domain decomposition based deep learning framework for nonlinear partial differential equations, Commun. Comput. Phys. 28, 2002 (2020).
  43. B. Moseley, A. Markham, and T. Nissen-Meyer, Finite basis physics-informed neural networks (FBPINNs): A scalable domain decomposition approach for solving differential equations, Adv. Comput. Math. 49, 62 (2023).
  44. https://anonymous.4open.science/r/dcpinns-ef12704/.
  45. P. Carr and D. B. Madan, A note on sufficient conditions for no arbitrage, Finance Res. Lett. 2, 125 (2005).
  46. U. Ayachit, The ParaView Guide: A Parallel Visualization Application (Kitware, Inc., 2015).
  47. S. Singh and S. Mittal, Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes, J. Fluids Struct. 20, 1085 (2005).

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Phys. Rev. E 113, 055301– Published 4 May, 2026
  • Received 9 June 2025
  • Accepted 14 April 2026
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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