Improving PINNs By Algebraic Inclusion of Boundary and Initial Conditions

"AI for Science" aims to solve fundamental scientific problems using AI techniques. As most physical phenomena can be described as Partial Differential Equations (PDEs) , approximating their solutions using neural networks has evolved as a central component of scientific-ML. Physics-Informed Neural Networks (PINNs) is the general method that has evolved for this task but its training is well-known to be very unstable. In this work we explore the possibility of changing the model being trained from being just a neural network to being a non-linear transformation of it - one that algebraically includes the boundary/initial conditions. This reduces the number of terms in the loss function than the standard PINN losses. We demonstrate that our modification leads to significant performance gains across a range of benchmark tasks, in various dimensions and without having to tweak the training algorithm. Our conclusions are based on conducting hundreds of experiments, in the fully unsupervised setting, over multiple linear and non-linear PDEs set to exactly solvable scenarios, which lends to a concrete measurement of our performance gains in terms of order(s) of magnitude lower fractional errors being achieved, than by standard PINNs. The code accompanying this manuscript is publicly available at, https://github.com/MorganREN/Improving-PINNs-By-Algebraic-Inclusion-of-Boundary-and-Initial-Conditions

Regularized Gradient Clipping Provably Trains Wide and Deep Neural Networks

In this work, we instantiate a regularized form of the gradient clipping algorithm and prove that it can converge to the global minima of deep neural network loss functions provided that the net is of sufficient width. We present empirical evidence that our theoretically founded regularized gradient clipping algorithm is also competitive with the state-of-the-art deep-learning heuristics. Hence the algorithm presented here constitutes a new approach to rigorous deep learning. The modification we do to standard gradient clipping is designed to leverage the PL* condition, a variant of the Polyak-Lojasiewicz inequality which was recently proven to be true for various neural networks for any depth within a neighborhood of the initialisation.

Investigating the Ability of PINNs To Solve Burgers' PDE Near Finite-Time BlowUp

Physics Informed Neural Networks (PINNs) have been achieving ever newer feats of solving complicated PDEs numerically while offering an attractive trade-off between accuracy and speed of inference. A particularly challenging aspect of PDEs is that there exist simple PDEs which can evolve into singular solutions in finite time starting from smooth initial conditions. In recent times some striking experiments have suggested that PINNs might be good at even detecting such finite-time blow-ups. In this work, we embark on a program to investigate this stability of PINNs from a rigorous theoretical viewpoint. Firstly, we derive generalization bounds for PINNs for Burgers' PDE, in arbitrary dimensions, under conditions that allow for a finite-time blow-up. Then we demonstrate via experiments that our bounds are significantly correlated to the $\ell_2$-distance of the neurally found surrogate from the true blow-up solution, when computed on sequences of PDEs that are getting increasingly close to a blow-up.

LIPEx -- Locally Interpretable Probabilistic Explanations -- To Look Beyond The True Class

In this work, we instantiate a novel perturbation-based multi-class explanation framework, LIPEx (Locally Interpretable Probabilistic Explanation). We demonstrate that LIPEx not only locally replicates the probability distributions output by the widely used complex classification models but also provides insight into how every feature deemed to be important affects the prediction probability for each of the possible classes. We achieve this by defining the explanation as a matrix obtained via regression with respect to the Hellinger distance in the space of probability distributions. Ablation tests on text and image data, show that LIPEx-guided removal of important features from the data causes more change in predictions for the underlying model than similar tests on other saliency-based or feature importance-based XAI methods. It is also shown that compared to LIME, LIPEx is much more data efficient in terms of the number of perturbations needed for reliable evaluation of the explanation.

Global Convergence of SGD For Logistic Loss on Two Layer Neural Nets

In this note, we demonstrate a first-of-its-kind provable convergence of SGD to the global minima of appropriately regularized logistic empirical risk of depth $2$ nets -- for arbitrary data and with any number of gates with adequately smooth and bounded activations like sigmoid and tanh. We also prove an exponentially fast convergence rate for continuous time SGD that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence of Frobenius norm regularized logistic loss functions on constant-sized neural nets which are "Villani functions" and thus be able to build on recent progress with analyzing SGD on such objectives.

Size Lowerbounds for Deep Operator Networks

Deep Operator Networks are an increasingly popular paradigm for solving regression in infinite dimensions and hence solve families of PDEs in one shot. In this work, we aim to establish a first-of-its-kind data-dependent lowerbound on the size of DeepONets required for them to be able to reduce empirical error on noisy data. In particular, we show that for low training errors to be obtained on $n$ data points it is necessary that the common output dimension of the branch and the trunk net be scaling as $\Omega \left ( {\sqrt{n}} \right )$. This inspires our experiments with DeepONets solving the advection-diffusion-reaction PDE, where we demonstrate the possibility that at a fixed model size, to leverage increase in this common output dimension and get monotonic lowering of training error, the size of the training data might necessarily need to scale quadratically with it.

Global Convergence of SGD On Two Layer Neural Nets

In this note we demonstrate provable convergence of SGD to the global minima of appropriately regularized $\ell_2-$empirical risk of depth $2$ nets -- for arbitrary data and with any number of gates, if they are using adequately smooth and bounded activations like sigmoid and tanh. We build on the results in [1] and leverage a constant amount of Frobenius norm regularization on the weights, along with sampling of the initial weights from an appropriate distribution. We also give a continuous time SGD convergence result that also applies to smooth unbounded activations like SoftPlus. Our key idea is to show the existence loss functions on constant sized neural nets which are "Villani Functions". [1] Bin Shi, Weijie J. Su, and Michael I. Jordan. On learning rates and schr\"odinger operators, 2020. arXiv:2004.06977

Capacity Bounds for the DeepONet Method of Solving Differential Equations

In recent times machine learning methods have made significant advances in becoming a useful tool for analyzing physical systems. A particularly active area in this theme has been "physics informed machine learning" [1] which focuses on using neural nets for numerically solving differential equations. Among all the proposals for solving differential equations using deep-learning, in this paper we aim to advance the theory of generalization error for DeepONets - which is unique among all the available ideas because of its particularly intriguing structure of having an inner-product of two neural nets. Our key contribution is to give a bound on the Rademacher complexity for a large class of DeepONets. Our bound does not explicitly scale with the number of parameters of the nets involved and is thus a step towards explaining the efficacy of overparameterized DeepONets. Additionally, a capacity bound such as ours suggests a novel regularizer on the neural net weights that can help in training DeepONets - irrespective of the differential equation being solved. [1] G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-informed machine learning. Nature Reviews Physics, 2021.

An Empirical Study of the Occurrence of Heavy-Tails in Training a ReLU Gate

A particular direction of recent advance about stochastic deep-learning algorithms has been about uncovering a rather mysterious heavy-tailed nature of the stationary distribution of these algorithms, even when the data distribution is not so. Moreover, the heavy-tail index is known to show interesting dependence on the input dimension of the net, the mini-batch size and the step size of the algorithm. In this short note, we undertake an experimental study of this index for S.G.D. while training a $\relu$ gate (in the realizable and in the binary classification setup) and for a variant of S.G.D. that was proven in Karmakar and Mukherjee (2022) for ReLU realizable data. From our experiments we conjecture that these two algorithms have similar heavy-tail behaviour on any data where the latter can be proven to converge. Secondly, we demonstrate that the heavy-tail index of the late time iterates in this model scenario has strikingly different properties than either what has been proven for linear hypothesis classes or what has been previously demonstrated for large nets.

Investigating the locality of neural network training dynamics

A fundamental quest in the theory of deep-learning is to understand the properties of the trajectories in the weight space that a learning algorithm takes. One such property that had very recently been isolated is that of "local elasticity" ($S_{\rm rel}$), which quantifies the propagation of influence of a sampled data point on the prediction at another data point. In this work, we perform a comprehensive study of local elasticity by providing new theoretical insights and more careful empirical evidence of this property in a variety of settings. Firstly, specific to the classification setting, we suggest a new definition of the original idea of $S_{\rm rel}$. Via experiments on state-of-the-art neural networks training on SVHN, CIFAR-10 and CIFAR-100 we demonstrate how our new $S_{\rm rel}$ detects the property of the weight updates preferring to make changes in predictions within the same class of the sampled data. Next, we demonstrate via examples of neural nets doing regression that the original $S_{\rm rel}$ reveals a $2-$phase behaviour: that their training proceeds via an initial elastic phase when $S_{\rm rel}$ changes rapidly and an eventual inelastic phase when $S_{\rm rel}$ remains large. Lastly, we give multiple examples of learning via gradient flows for which one can get a closed-form expression of the original $S_{\rm rel}$ function. By studying the plots of these derived formulas we given a theoretical demonstration of some of the experimentally detected properties of $S_{\rm rel}$ in the regression setting.